affdl-tr-76-55 volume i

AFFDL-TR-76-55 Volume I

AERODYNAMIC STABILITY TECHNOLOGY FOR MANEWERABLE MISSILES

Volume I. Configuration Aerodynamic Characteristics

MAR TIN MARIETTA CORPORA TION, ,

ORLANDO DIVISION c

P. 0. BOX 5837 ORLANDO, FLI-RIDA 32805 .

.- . .F-- -

MARCH 1979

TECHNICAL REPORT AFFDL-TR-76-55, 'Vol. 1 Final Report for period February 1975 - December 1976

I Approved for public release; distr~cution unlimited.

AIR FORCE FLIGHT DYNAMICS LABORATORY AIR FORCE WRIGHT AERONAUTICAL LABORATORIES

0

AIR FORCE SYSTEMS COMMAND WRIGHT-PATTERSON AIR FORCE BASE, OHIO 4543 3

Reproduced From Best Available Copy

When Government drawings, specifications, or other data are used for any pur- pose other dun i n mmect fon with a defini tely related Covernknt procurement aperation, the United States Government thereby incurs no responsibility nor any obligation whatsoever; and the f a c t thd t the government may have formulated,. furnished, or i n any way supplied the said drawi ngs, specifications, or other data, i s not to be regarded by implication or otherwise as i n any munner licen- sing the holder or any other person or corporation, or conveying any rights or permission to manufacture, use, or sell any patented invention that m y i n any way be related thereto.

This report has been reviewed by the Information Of f ice (OI) and i s releasable to the Nationa: Te:Lul~cdl Informtfon Service (NTIS) . lit NTIS, i t rill be-available to the general public, including foreign nations.

Thls technical report has heen reviewed and i s approved for publlcation.

W. H. LANE R. 0. 'ANDERSON, Chief '

Project Engineer ' Con t rol Dynamics Branch Control Dynamics Branch F l f ght Control Division

FOR THE COMMANDER I

Air ~ b r r e Flight Dynamics Laboratory

"ff your address has changed, if you wish to he removed from oB1r mailing l i s t , or i f the addressee is no lrmger employed by your organization please notify

AFFVL/FGC ,W-PAFB, OH 45433 to help us maintain a current mailing l i s t " .

Copies o f this report should not be retu.rne3 unless retura i s required b y se- curf t~ considerations, contractual obl i ya t i ons, or notice on a s p e d f i e document. AIR FORCUS6780121 Mw 1979 - 33

UNCLASSIFIED

1 i Final ,Kaput. F e b m krodytumic Stability Tec ology for ~ e u v e r a b l e Missiles. dl . I. Con- 1, R 7 5 * De-76 -, f i g u r n : i o : i A e r o d y n ~ ~ m l c ~ ; ~ m c t e ~ l s t f c ~ , ~ f

D C C R F O ~ M I U G ORGAHIZATION NAME AND AODRCSS 10 PROGIIAM CLLMCNT PROJECT. TASK AREA 6 WORK UNIT NUMBER$

Martin Marietta Corporation \ . 1 Orlando Division, PO Box 5837 Orlando, FL 32805

/

h REPORT D A T E I CONTROLLIHG OFF ICE NAMC AND ADDRESS

U.S. Air Force Flight Dynamics Laboratory - Wright-Patterron Air Force Bas&

I, 6 M S T R I ~ U T I O N STATEMENT fof thlo R.pnrf)

Approved for public release; dirtribution unlimited

---- - - -- -"- - ----- - -- - a n $ ~ r ) & ( r ( ~ n t l r . r . ~ r r r e t r r w aid. If n*c r..ar, u r d Isknttl! hv Dtoa h n w d w l

hie etudy developed empirical method. to predict aerodynamic characteriaticr f body-tail, body-wing-tafl and body-etrake-tail miseile configurations. ethods cover the Mach number range from 0.6 to 3.0. Methods covrr rhr; indi- idual body and tail characteristics at angles of attack from 0 co 180 degrees. or vinged bodies the methods encompass angles of attack up to about 30 degreee. 11 mutual interference effects are accounted for, alloving accurate prediction f force and wment coefficients. -

FOREWORD

This r e p o r t was prepared f o r t h e U. S. A i r Force F l i g h t W a c s

Laboratory, Wright-Patterson Air Force Base, Dayton, Ohio, under c o n t r a c t

number F33615-75-C-3052 as p a r t of P r o j e c t 8219. The work was performed

st t h e Orlando Divis ion of Martin Mar ie t t a Aerospace i n Orlando, Flor ida .

The repor ted e f f o r t began i n February 1975 and ended with t h e s u b m i t t a l

of t h e d r a f t of t h i s f i n a l r e p o r t i n December '1976.

The p r i n c i p a l i n v e s t i g a t o r s were J. E. F i d l e r and G. F. Aie.'k?. The

t e c h n i c a l monitors f o r t h e F l i g h t Dynamics Laboratory were Dr. Robert Nelson,

Lt William Miklos and Mr. William Lane.

The a u t h o r s wish t o express t h e i r g r a t i t u d e t o t h e aforementioned

coh t rkc t monitors f o r t h e i r guidance and support and recognize a s p e c i a l

debt t o MI Lane f o r h i s e x t r a o r d i n a r y e f f o r t i n reviewing t h i s r e p o r t and

t h e s i g n i f i c a n t c o n t r i b u t i o n towards t h e r e a d a b i l i t y and o v e r a l l q u a l i t y o f

t h e repor t . The a u t h o r s would a l s o l i k e t o exp iess t h e i r g r a t i t u d e t=

Kr. Will iam Baker, Arnold Engineering Development Center, f o r h i s cooperat ion

i n providing easy access t o t h e 180 degree, body p l u s t a i l d a t a bank. Many

s i n c e r e thanks a r e due t h e fol lowing a s s o c i a t e s a t the Martin Mar ie t t a , Orlando

Division: G. S. Logan; Jr., D. T,. Moore and R. L. Swann.

Accession For .)

mc TAB Unannouncell Justification

-- ?. ' . ' - &

TABLE OF CONTENTS

1.0 Introduction . . . . . . . . . . . . 1 . . . . . . . . . . . . 2.0 Experimental Data Sources and Modela . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . 3.0 Aerodynamic Data Trends

. . . . . 4.0 FQrmulation of the Aerodynamic Prediction Equations I '

. . . . . . . . . . . . . . . . . . . . . 5.0 .Aerodynamic Methods

5.1 Isolated Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.i body Normal Force

5.1.2 Body Center of Pressure . . . . . . . . . . . . . 5.1.3 Body Axial Force . . . . . . . . . . . . . . . . . . 5.1.4 Fin Normal Force . . . . . . . . . . . . . . . . . 5.1.5 Chordwise Center of Pressure . . . . . . . . . . .

5.2 Body-Tail Configurationo . . . . . . . . . . . . . . . . 5.2.1 Tail-on-Body Nonnal Force . . . . . . . . . . . . 5.2.2 Tail-to-Body Carry-over Normal Force . . . . . . . 5.2.3 Tail-to-Body Crrry-over Normal Force

Center of Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Body-Strake-Tail Configurations

. . . . . 5.3.1 Incremental Normal Force Due to Strakes '

5.3.2 Center of Pressure for Incremental Normal . . . . . . . . . . . . . . . Force Due to Strakes

5.3.3 Incremental Normal Force Due to Tails . . . . . . 5.3.4 Center of Pressure for Incremental Normal . . . . . . . . . . . . . . . . Force Due to Tails

TABLE OF CONTENTS (Concluded)

Page

. . . . . . . . . . . . . . 5.4 ,Body-Wing-Tail Configurations 259

. . . . . . 5.4.1 Incremental Nonnal Force Due t o Wings 259

. 5.4.2 Effective Ccnter of Pressure f a r Incremental . . . . . . . . . . . . . . . . Force Due t o Wings 274

5.4.3 Ta i l Incremental Normal Force Due t o Wing . . . . . . . . . . . . . . . . Vortex Interference 289

5.4.4 Effective Center of Pressure of the Incremental . . . . . . . . . . Tai l Normal Force Due t o Wings 306 . . . . . . . . . . . . . . . 5.5 Thrust Vector Control Effects 310

5.5.1 Incremental Body ~ o r m a l Force Due to , P lum . . . . . . . . . . . . . . . . . . . . . Effects , 310 %

5.5.2 Effect ive Center of Pressure f o r Incremental Body. . . . . . . . . Normal Force Due t o Plume Effects 323

5.5.3 ~ncremental ' .Tail Normal Force Due t o Plume . . . . . . . . . . . . . . . . . . . . . Effects 334

5.5.4 Effective Center of Pressure of Incremental Ta i l Normal Force Due t o Plume Effects . . . . . . . . 351

6.0 Conclusions and Recomsndations . . . . . . . . . . . . . . . a 356 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.0 References 358

LIST OF ILLUSTRATIONS

Figure PagL

l a Methodology Requirements f o r TVC Missiles . . . . . . . . . 5

l b Methodology -+irementa fo r Aerodynamically C o n , t r o l l e d H f e e i l e e . . . . . . . . . . . . . . . ~ . . . . . 6

2 Schematic of Total Data Base . . . . . . . . . . . . . . . . 9

3a Martin Marietta Main Body Model i n the NSRDC 7 ' X 10' Transonic Tunnel a t Sixty Degreee Angle of Attack , 10

3b Martin Marietta Ta i l Models . . . . . . . . . . . . . . . 11 4 Vortices Produced by the Reattachment of Lower Surface . . . . . . . . . . . . . . . . . . . . . . BoundaryLayer 13

5a Fin Normel Force Coefficient (H-0.8, Aspect Ratio . . . . . . . . . . . . . . . . . . . . . . . . . ~ f f e c t s ) 19

5b Fin Chordwise Center of Preeeure (M-0.8, Aspect . . . . . . . . . . . . . . . . . . . . . . R a t i ~ E f f e ~ t r ) 20

5c Fin Normal Force Coefficient (M-2.0, Aapect Ratio . . . . . . . . . . . . . . . . . . . . . . . . . Effects) 21

5d Pin Chordwise Center of Pressure :M-2.0, Aspect . . . . . . . . . . . . . . . . . . . . . . R a t i o E f f e ~ t s ) 2%

6a Pin Normal Force Coefficient (M-0.8, Taper Ratio . . . . . . . . . . . . . . . . . . . . . . . . . ~ f f e c t a ) 23

6b Fin Chordwiee Center of Preeeure (W0.8, Taper Ratio . . . . . . . . . . . . . . . . . . . . . . . . . ~ f f e c t m ) 24

6c Fin Normal Force Coefficient (M-2.0, Taper Ratio . . . . . . . . . . . . . . . . , Effecte) . . . . . . . . . . :. 25

6d Fin ~hordwise Center of Preeaure (M-2.0, Taper Ratio ~ f f e c t a ) . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7b Fin Chcrdwise 'Center of Prerrure (Mach Effect.) . . . . . . 28

8a ,Variat ion of Induced Out-of-Plane Forces and Hmntr (M-Q.6). . . . . . . . . . . . . . . . . . . . . . 29

,LIST OF ILLUSTRATIONS (Cont 'd)

Page

Variation of Indcced Out-of-Plane Forces and Moments (M-2.0). . . . . . . . . '. . . . , . . . . . . . . . 30

Out-of-Plane Forces and Moments Due t o Vortex . . . . . . . . . . Asyrumetry (AR = 0.5, 1\ 1.0, d/8 0.5) 31

Out-of -Plane F ~ r c e s and Moments Due t o Vortex Asymmetry (AR = 0.5, X = 0, a!s - 0.4). 32 . . . . . . . . . . Comparisbn of Ta i l Normal Farces . . . . . ., . . . , . . . . . 33

Comparison of Rolling Momentti. . . . . . . . . . . . . . . . 34

Comparison of Experimental ond Predicted Results (CN 1, Mach - 0.6. . . . . , . . . . , . . . . . . . . . . . . 48

B comparison of P,x2erlmenta?. and Predicted ~ b u l t s ( C N ) , M a c h m 1 . 1 5 . . . . . . . . . , . . , . . . . . , . 4 8

B - .Comparison of Eiperimental and Predicted Results

1, Mach = 1.30 . . . . . . . . . . . . . . . . . . . . 49

Comparison of Experinrental and Fredicted Results (c&. W . C ~ - 2.6 . . . . . . . . . . . . . . . . . . . 49

B Coefficients for Calculation of C (A1) . . . . . . . . . 50

N~ Coefficients for Calculation of C (5) . . . . . . . . . 50

N~ Curves for Transonic C*, ( tN/d - 1.5). . . . . . . . . . . . 51

a Curves fo r Traneonlc CN ( tNld = 2.5). . . . . . . . . . . . 52

a Curves fo r Transonic CN (LNld - 3.5). . . . . . . . . . . . 52

a Curves f a r Superrronic C ( tN/d = 2.5). . . . . . . . . . . . 53

Curvea for Supersonic $ (rN/d = 3.0). . . . . . . . . . . . 53 a

Curves fo r Supersonic C ( tN/d - 3.5). . . . . . . . . . . . 54

Curves f o r Supersonic C (I. Id = 4.0). . . . . . . . . . . . 54 *a . . . . . . . . . . . . Correlation Factor for End E f f e c t s , 55

Variation of 0 with Mach Number . . . . . . . . . . . . . . 55

Curves For Detetmining Transonic Values of n * . ~ . . . . 56

Figure

22 a

22b

23

LIST OF ILLUSTRATIONS (Cont'd)

Basic Valuer of C * . . * . . . . . . . . . . . . . . . . 57 dc

Croesf low Drag Coefficient (Subcritical Crossflov, M q O . 4 ) . 57 c-

Comparison of Experimental and Predicted Results (C ),Hach-0.6.. . . . . . . . . . . . . ' . . . . . a . 57 N~

Comparison of Experimental and Predicted Resu,lts (C ), Mach -1.15 . . . . . . . . . . .' . . . . . . . . . . . 58 N~

Comparison of Experimental and Predicted Results (C ),Mach.1.30 . . . . . . . . . . . . . . . . . . . . '59 N~

Comparison of Experimental and Predicted Results

Comparison of Experimental and Predicted Results (C ),Mach-2.86 . . . . . . . . . . . . . . . . . . . . 60 N~

Comparison of ,Experimental and Predicted Results (5 ). Mach = 0.85, 1.20, and 2.25. . . . . . . . . . . . . 60

5 Transonic Tangent Ogive-cylinder Zero Angle of Attack Centers of Pressure (Li /d - 3.5) . . . . . . # . . . . 70

Transonic Tangent Ogive-Cylinder Zero Angle of Attack Centere of Pressure (L Id - 2 . 5 ) . . . . . . . . . '. 70 N

Transonic Tangent Ogive-Cylinder Zero Angle of Attack Centers of Pressure (iN/d' - 1.5) . . . ; . . . . . . 7 f l

Supersonic Tangent Ogive Cylinder Zero Angle of Attack Center8 of Pressure (iN/d - 4.0) . . . . . . . . . . 71 Supersonic Tangent Ogive - Cylinder Zero Angle of Attack ,Centerr of PressuLa ( LN/d * 3.5) . . . . . . . . . . 71 Supersonic -:ngent Cgive - Cylinder Zero Angle of Attack Centers of Pressure ( $ I d = 2.5) . . . . . . ,, . . . 7 1

Increment' in Center of Pressure Between Angles'of Attack of 0 and 20 degrees . . . . . . . . . . . . . . . . i 2

-9 '7 Polynomial Coefficients, Low Angle of Attack . . . . . . . 4 , .

Polynmial Coefficients . High Angle bf. Attack. . . . . . . 73


Figure ,

34 Comparfaone Between Predictions and E x p e r h n t a l

Conpariaons Between :.=dictions and Experimantal

3 6 Conparisone Between Predictions and Experimental

37 Cornparisone Between Prediction8 and Experimental

38 Compar ieons Between Predict ions and Experimental Data kP . Hach - 3.0 . . . . . . . . . . . . . . . . . . 76

--B d

39 Vdriatiori with Mach Number of 180-Degree Axial Force C o e f f i c i e n t . . . . . . . . . . . . . . . . . . . . . . . . 84

40 Cornpariaon Between Predicted and Experimental C . . . . . . . . . . . . . . . . . . . . . . (a-f (Traneo nic) 85

41a Curves for Determining CA (LN/d - 1.5). . . . . . . : . . 87

l b

41b , Curves fo r Determining CA (tN/d - 2.5). . . . , , . . . . . 87

'2,

41c Curvee fo r Determining CA (tN/d = 3.5) . . '. 88 . . . . . . . lb

4 2 Scaling Factor fo r C . . . . . . , . . . . . . . . . . . .88

4 3 Variation of EA with Mach Number . . . . . . . . . . . . . 89

4 4 Basic Curves of f ( ~ , a) Calculated from Power Seriee . . . 89


Figure

45 ohp par is on Between Predicted and Experimental Daf:a '

C (Shtpersonic) . . . . . . . . . . . . . . . . . . . . . . . 90 % Power Series Parmeters for Equation (24) . . . . . . . 104 Lift Curve Slope for Taper Ratios 0-1.0 . . . . . . . . . 105

. . . . . . . . . . Variation of C (+/2) with b c h 'Number 107 N~

a', Angle of Attack Above Which ACN Must be Applied (Subsonic only) . . . . . . . . . . . . . . . . . . . . . . . 108 . Dimensionless C Increment Above a,' . . . . . . . . . . . . . 109

N

Haxiinurn Increment of 'Normal Force Above a' Ac% * (Subsonic Only) . . . . . . . . . . . . . . . . . . . . . . . . 110

Comparison of Predic~ed and Experimental C , Mach -0.8. 110 N~

Comparison of Predicted and Experi~ental C Mach - 0.98 111 N- *

1

Comparison of Predicted and Experimental ,C , Mach -i-02 . 111 N~

V~rfation of Fin Normal Force at a - 90' with Mach'No. . . . . . . . . . . . . . . . . . . . . . . . . . 112 Variation of Normal Force Coefficient, C (30). with b c h ~ o . , a - 30. (A I 0) . . . . . . . NTe . . . . . . . . 113 Variation of Normal Force Coefficient, C (30). with . . . . . . . . . . . . . Maih No., ,a - 30' (A - . 5 ) . N ~ . '. 113 Variation of Norms1 Force Coefficient, C (30). with . . . . . . . . . . . . . Mach No., o - 30. (A = 1.0) N ~ . ; 113

Variation of C (30) wtthMach NuPzber . . . . . . . . . . 114 %a

Parer Series Parametere for Equation (26) . . . . . . . . . 115 Comparisori of Predicted and Experimental from 30 to 30 degrees . . . . . . . . . . ".T . . . . . ,. . 116 Curves for Modifying tN Method, (A - 0, AR - 1.0, Subsonic) . . . . . . . . . . . . . . . . . . . . . . . . . 116


Page

An Exanple Using AC . . . . . . . . . . . . . . . . . . . 116 N?:

Comparison of Method and Test ,C (A = 0, AR = 0.5) . . . 117 N~

Comparison of Method and t e s t , ( X = O . ~ , A R I ~ . 5;X=0,ARm1*0: 118 5,

Comparison of Test t o Methods to 180'. M = 0.6 (CN_). '. A19 1

~dmpa t i son of Test and ~ e t h o d ,. M = 2.0 (C ) . . . . . .' 120 N~

'comparison of Test and Method, M = 7.5 (C . . . . . * . 120 N~

Comparison of Test and Method, M = 3.0 (C )(~=1.0,AR-l.o) 121 N~

Comparison of Test and Method, M = 3.0 (C )(a= 0 ~ b 1 - 0 ) 121 N,

1

Chordwise Center of Pressure Variat ion t o 180 Degrees . . . . . . . . . . . . . . . . . . . . . . . . . 136

Chordwfse Center of Pressure Var ia t ion with . . . . . . . . . . . Taper Ratio a t Alpha of 90 Degrees 136

X c ~ Basic Curves, f o r - a t Reference p c h ~ u k b e r 0.98 (0-180Degrees, 'R A R = 0 . 5 ) . . . . . . . .,. . . . . 137

X c ~ Basic Curves f o r - at Reference Mach Number 0.98 (0-180 degrees, 'R 'AR = 1.0). . . . . . . . . . . . . 137

X c ~ Basic Curves f o r - a t ~ e f e r e n c e Mach Number 0.98 (0-180 Degrees, C~ AR = 2.0). . . . . . . . . . . . .' 1?7

X c ~ Basic curve; f o r - a t Reference Angle of Attack . . . . 175-180 Degrees 'R (M = 0.6. Oo 3 .O, AR = 0.5) 138

Basic Curves f o r k a t Reference Angle of Attack 175-1e0 Degrees L~ (M = 0.6 t o 3.0, AR = 1.0) . . . . 1 3 p

X c ~ Baatc,Curves f o r 7;- a t Reference Angle of Attack

b 1 7 ~ 1 8 0 Degrees R !M = 0.6 t o 3.0, AR = 2 . 0 ) . . . . 138

Power Se r i e s C o n s t a ~ t s Versus Angle o f ' h t t a c k . . . <;9

Mach Number Correction Fector f o r a< 90 Degrees , . . , -43

Variat ion of A ~ ( X ~ ~ / C ~ ) with Mach Number a t Alpha of 16C Degrees . . . . . . . . . . . . . . . . . . I h Q

Comparison of Predicted and Experimental cen ter of Pressure Location, X . M 1.15 . . . . . . . . . . . 141

CPT - C~

' Figure

7 7

LIST OF ILLUSTRATIONS (Cont 'd) '

Page

Comparison of Predicted and Experimental C.P. Locat ion . . . . . . . . . . . . . . 141

Comparison of Predicted and Experimental C.,P. Location, X& M - 1 . 3 . . . . . . . 142 . . . . . . . . . . T

q ( B ) Ratio a t Zero Angle of Attr.c'r . . . . . . . 150

General Coefficients for Calculation of ( I ( A o . 151

General Coefficiente f o r Calculation of (\) . . 152

General Coefficients for Calculation of Ry( ( ) . . . 153 B) A2

Idterference actor a t Angle of ~ t t a c k ,of 90 Degrees . . 154

Comparison of Experimental and Predicted Results,

Comparison of Experimental and Predicted Resulte, M - 2 . 0 . . . . . . . . . . . . . . . . . . . . 157

Comparison of Experimental and Predicted Results, , M - 3 . 0 . . . . . . . . . . . . . . . . . . . . 158

Comparison of Experimental and Predicted Results, . . . ~ r 1 . 1 5 ; . . . . . . . . . . . . . - 8 . . 159

Comparison of Experisental and Predicted Results, C r M - 0 . 8 . . . . . . . . . . . . . . . . . . . . . 160 N ~ ( ~ )

Transonic I B(T)'

Schematic . . . . . . . . . , . . 166

Curves for Estimation of Traneohic I ( a l l A ) . . . . . 167 a

x i i i

Figure

91b

91c

92

93

94.

94b

94c

LIST OF ILLUSTRATIONS (Cont 'd)

Curvea f o r E o t h t i o n of Transonic Ic ( a l l A and M) . . . . . . . . . . . . . . . . . . . . . . 167

, Comparison between Predicted and Experimental

Curves f o r Estimation of Supersonic 11 . . . . . . . . . 169

. . . . . . . . . . Curves fo r Estirrmtion of Supersonic I2 169

. . . . . . . . . Curves f o r E s t i ~ t i o n of Supersonic I3 169

. . . Cornpariaon Betveen Predicted and Experimental I BCr)

170

Curves f o r Determining X with Afterbodied C P ~ (TI CR

f o r Supersonic Speeds . . . . . . . . 1 8 3

Curves f o r Determining XCp f o r No Afterbodiee a t - B(T) 5, ~u&!rronic ' Speeds . . . . . . . . . . . . . . . . 184

Curves fo r Determining XCp fo r Subsonic - B(T) 5, . . . . . . . . . . . . Speeds (Zera Leading Edge sweep) 185

Curvas for Determining XCp f o r Subsonic - B(T) CR Speeds (Zero railing Edge Sweep) . . . . . . . . . . . . 186

Coefficient6 Required f o r Evaluation of

XCP . . . . . . . . . . . . . . . . . . - B ( T ) . ... . ' a - 1 8 7 cR Comparison Between Predicted and Experimental Data


Figure EE

102 Comparison Between Predicted and Experimental Data, X . . . . . . . . . . . . . . . . . . . . . . . I 8 9

"ST

ACN General Curve Form . . . . . . . . . . . . . . . . 196 BS

Coefficients for Calculating ACN . . . . . . . . . , . . . 198 BS

comparison of Test Data and Method,

General Curve Form, XCp . . . . . . . . . . . . . . . 211 AES

Polynomial Coefficients for Calculating X 213 =bBs

J and K Values far Calculating XCp . . . . . . . . . . 216 ABS

Comparison of Teat Data and Method, XCp /d . . . . . . . 217 ABS

Comparison of Test Datr! and Method, XCp /d . . . . . . . 219 BS

Coefficients for Calculation of AC (A1) . . . . . ,. 227 " ST

Coefficient. for Calculation of ACN (A,] . . . 228 ' BST

Coefficients for Cslculation of AC (A3) . . . . . . 229 N~~~

Kp(g) and R~tios (Slender Body Theory)., 230 . . . . . . Comparisons of Predicted Results with Experimental Data, A . . . . . . . . . . . . . . . . . . . . 231

'BST

Figure

118

119

120

121

LIST OF ILLUSTRATIONS (Cont ' d)

paac

l(rO) and K Y t i o r (Slender Body Theory) . . . . . . 245 B(T)

. . . Tai l Alone Center of Pressure a t Subsonic Spe9de. 246

Tail Alone Center of Pressure a t Supersonic S p e e d s . . . . . . . . . . . . . . . . . . . . * . . . . 247

X B(T) (or Subsonic Speeds Cutvar fo r Determining CP

C~ . . . . . . . . . . . . . . . (Zero Trai l ing Edge Sweep) 248

X Curves f o r Determining CPB a fo r No Af terhody -

, , C~

. . . . . . . . . . . . . . . . . . a t Supersonic Speed? 249

Curves f o r ~ a t e d n i n ~ 'CP Bo for Subsonic Speeds

C~ (Zero Leadihp Edge Sweep) . . . . . . . . . . . . . . . . 250

X Curves fo r Determingt cp with Afterbodiem a t -m . . . . . . . . . . . . . . Superronic Spaedr I CR ' 251

Coefficients f o r Calculation of XCp, (A1) . . . . . . . . 252 - C~

Coefficients for C a l c u l ~ t i o n of XCp, (A2) . . . . . . . 253 - C~

Coefficient8 f o r Calculat ion of XCp, (Aj) . . . . . . . 254 -

Comparison Between Predicted and Experimental

Reaults,XCp / d , k O . 6 . . . . . . . . . . . . . . . . ' . 355 BST

Comparison Between Predicted and Experiment81

Rerultr , XCp /d, M-0.85 . . . . . . . . . . . . . . . 256 BST ,

x v i

Figure

130

LIST OF ILLUSTRATIONS (Cont 'd)

Comparison Between Predicted and Experimental Resultr, XCp /d,l+1.2.. . . . . . . . . . . . . . . . 257

BST

Cornpar: son Between Predicted and Experlmental Results, XCp Id, l+ 2.2 . . . . . . . . . . . . . . . . 258

BST

Comparisons of Existing Method Predictions with Experimental Data, AC . . . . . . . . . . . . . . . .266

N~~

Ratfo at Zero Angle of Attack . . . . . . . . . . . 267 Comparison Between Predictei-j and Experimental Rerulta, bCN . Configuration 2, W1.1. . . . . . . . . . 268

BW

Configurations (Body + Wing) . . . . . . . . . . . . . ., . 269

Comparisons Between Experimental and Predicted Results, AC . Configuiationcl 1 and 3, Ml. 1 . . . . . . 270

NBw

Comparisons Betveen Experimental and Predicted Rcsults,AC ,p3.08.. . . . . . . . . . . . . . . . . 271

Ng-,

Comparisons ,Between Experimental and Predicted Results, ACN ~ M 1 . 9 . . . . . . . . . . . . . . . . . . 272

B W

Comparison Between Experimental and Predicted ResulLs, AC . l+ 0.85 . . . . . . . . . . . . . . . . . 273

N~~

. . . . . . %(B) and %(w) btios (Slender Body Theory). 279

Wing Alone Center of Pressure at Subsonic Speeds. . , . , 280 . . . Wing Alone Center of Presrure at Suprrsanjc Speeds. 281

Curve6 for Determining XCp /cR at Subrontc Speeda. . . 282 B (W

Curver for Determining XCp /CR with Mterbody e (W . . . . . 283

' x v i i

LIST OF ILLUSTRATIONS (Cont ' d)

. . . . . . . . . . . . . . . Conf igura t ion8 (Body + Wing) 284

Compariron Between Prediction. and Experimental . . , . . . . . . . . . . . . . . Data, ItCP Id, ~ 0 . 8 5 ' 285 ' ABW

Comparicron Between Predictionr and Experimental . . . . . . . . . . . . . . . . . . Data, XCP I d , W1.1 286 AB W

Compar iron Between Predict ions and Experimental . . . . . . . . . . . . . . . . . . Data, XCP Id, W1.9 , 287 BW

Campariaon Between predict ions and Experimental . . . . . . . . . . . . . . . . Data, X / d , W 2 . 8 6 . ., 288 CPdw ' I

Transonic Wind Tunnel Test Configurations . . . . . . . . 299

. . . . . . . . . . . . . . . . . . Wing Vortex Location 300

. . . . . . . . Wing Vorcex Induced' Ta i l Angle of Attack. 301

Compariron Between Predicted and Experiwntal . . . . . . . . . . . . . . . . . . Results, AC 1 . 1 302 NTwv

Comparison Between Predicted and Experimental . . . . . . . . . . . . . . . . . . . . . R e s u l t , c W0.7 303 BUT

Comparison Between Predicted and Experimental R c ~ u l t n , C , W0.85 . . . . . . . . . . . . . * . . . 304

N ~ U T

Cobpariron Between Predicted and Experimental . . . . . . . . . . . . . . . . . R e a ~ l t 8 . C ~ , W 2 . M 305 BW

Comparison Between Predicted and Ex?erhental . . . . . . . . . . . . . . . . Results, X Id, W0.85 308 "BW

Comparison Between Predicted and Experfmental . . . . . . . . . . . . . . Resulte, X / d , + 2 . 3 6 . . 309 CPBwr

Fibre

159

160

161

162 (a*)

163 (a*)

164 (8-4

165

166

167.

167b

168 ( a 4

LIST OY ILLUSTRATIONS (concluded)

General Curve Form, ACN . . . . . . . . . . . . . . . . 317 BP

Power Serier A for Calculating A% . . . '. . . . . . . . 318 BP

Amplification Factorr for Calculating ACN . . . . . . . 319 BP

Comparisons Between ~rcdictions and Experimental . . . . . . . . . . . . . . . . . . . . . . . Data, AC 320 , N~~

Co~parison of Body Alone XCp /d (Jet-On and Jet-Off) . . . 328

Comparison Between Predictions and Experimental

General Curve Forma, . . . . . . . . . . . . . . . 343 Amplification Factors for Calculating ACN '. . . . . 344

TP . . . . . . . . Pawer Sorier A for Calculatiq A% M51.2 346 , TP --

Pmmr Serieo A for Calcuhting ACN , 1.2-2.2 . 347 rP

Camparirons Between Predictions and Experime~tal . . . . . . . . . . . . . . . . . . . . . Data* A%T

, I. 348

Corparieon between Jet+ and Jet-Off Tail Centers . . . . . . . . . . . . . . ofpregeure . . . . . . . . U 3

x i x

LIST OF SYMBOLS

General coefficients

2 Aopect ratio, (2b) /S (2 panelr)

Strake aspect rat lo

Body radius * inches Polynominal expannion coefficients

General coef f i'cients

Exposed semispan * inchen Gcneril coef f icient s

Axial force coefficient

Axial force coefficient, omitiing base effects

Baric value of C A,

h i a l force coe~ficient due to base effec.tp

Drag coefficient

~ttching moment coefficient

Normal force coefficient, based on Sref

Body alone CN, jet-off and jet-on, respectively

Body + straker normal force coefficient

Total CN on the body in the presence of rtrakes and tailr, jet-off and jet-on.

Normal force coefficient at a = 90.

% * % Total 5 cf four atrake. i n presence of body, S(B)total (B)PtOtal jet-of f and j at-on

% Single tail p a e l alone normel focce coefficient

%(BP Total $ of four t a i l s i n presence of body, jet-off

t o t a l and jet-on

Single t a i l panel normal force coefficient i n the presence of a body, .jet-off and jet-on

Root, chord % inch88

Strake root, chard length % inches

Base stagnation pressure coefficient .

Body cross-sectional diameter * Inches

Nozzle ex i t diarhter % i n c h r ~

Amplification factor a t a = 55.. 110', and 160° - f(M)

Value of dC 1% a t a = 70. [=f (M) ] N~~

Value of AC 1% a t a - i45' [-f (M) 1 IP

Mach number correction used i n conjunction with

Vertical distance between wing vortex core and t a i l surface a t LS

Coefficients for transonic range of I B (TI

Coefficients for supersonic range of I B (TI

Strake contribution t o zarryover CN on body, jet-of f ind jet-on

Tail contribution t o carryover CN on body, jet- cff and jct-on

Total carryover due t o strakes md ' t a i l r , jet-off

J2 S a l e fac tor f o r XcpABS i t a-60'

hmplffication factor, peak value of ACNBS a t a - 57. and 1 3 5 .

5, Scala fac tor f o r mABS a t a-60'

Value of XCpB/dpep a t a - 120.

Ratio of normal force on t a i l i n the presence of body t o t a i l aloile normal force

Ratio of normal force on body due to t a i l s t o t a i l alone normal force

h t i o of normal farce on body due t o wings to wing alone normal force

Ratio of normal force on tha wlng i n tha presence of body co wlng alone normal force

Length (\, incher

Fineness r a t i o

Hi r r i l e t o t a l length ?, inchea

Length of missile cylindrical auction ?. incher

54 )iirrile nose length ?. incher

Dirtance batween wing t r a i l i n g edge-and t a i l leading edge a t a l a t e r a l distance YW

M Freestream Mach number

"k ~ e t momentum r a t i o - qJ/q,

M. 3. M i r d l e r t a t lon (incher from the aore)

N Norul force % lbf

LIST OF SYMBOLS (CONT'D)

Ta i l s&ispan, measured from body center l ine s incher

Jt:t dynamic pressure a t nozzle e x i t I, lbs/rq. f t .

Freestream dylumic p re swra % i b d r q . f t .

X Tangent ogive nose rad4,cl, o r value of mS a t a - 60. * inch.' - d

Reynolds number

Ta i l area r a t i o - S , Tt 'ref

Iaterference fac tor (C *TO)

Interference fac tor (

body r a d i w c inches

Radial d i r tance manured from vortex core inches

Area * rq. f t .

Mdy planforn area * rq. f t.

b f e r e n c e a r a r - & rq. f t . 4

Strake s ing l e span exposad area c rq. ft.,

Aret of two s t rakes + planfonu area of body between s t rakes T a i l s i ng l e panel exposed area * sq. f t .

,Wing mingle panel expored area * rq. f t . ,

Total t a i 1 rpan including body

Vortex tangential ve loc i ty a t a dis tance r

Axial d i s t ance * inches

Diptance t o center of planform area s incher

I.ocation of forward s t rake segment centroid r e l a t i ve t o LE

LIST OF SYMBOLS (CONT'D)

%P - A B C P C~

Location of a f t r t r ake segment centroid r e l a t i v e t o LE

Location of net s t r ake centroid r e l a t i v e t o ti:,

Center of pressure of carry-over loading on body measured from t a i l root chord leading edge

Center of pressure o f , t h e t a i l measured from the body nose

Chordwise cen t e t of pressure of t a i l i n the presence of a body ~ e a s u r e d from t a i l root chord leading edge

Body alone cen te r of pressure s t a t i o n , jet-off and j e i - o n , r e l a t i v e t o the nose

Body + strakeo center of pressure, r e l a t i v e t o the nore

Center of premrure of a bedy-r t rake-tai l c o m b i ~ t i o n , jet-off and jet-on

Ef f a c t i v e cen t a r of presrure (M. S.) of t o t a l carryover CN d ~ e t o s t r akes + t a i l s , jet-off and jet-on

Effec t ive cen ta r of preelr&e (H.S.) of r t r h carryover on body CN, jet-uff and jet-on

Effec t iva cen te r of pressure (M.S.) of t a i l carryover, on body CN, j e t -of f and Jet-on

Center of pressure of AC , r e l a t i v e t o ' s t rake LE N~~

Center o f ' pressure of LC , r e l a t i v e t o the nose N~~

Center of pressure of AC a s a percentage ~i *BST

root chord measured from the wing root chord leading edge

Cmte r of pressure of LK measurrd in diameters from the nose N~~

x x i v

Effect ive cen te r of preraure of A$ 'TUV

Chordwise cen te r of pressure (nondimeneionalited by panel root chord, CR)

Center of pressure a t a - i degrees

a T a i l cen te r of preeeure at at = 160. for bas ic h c h - 0.98

Ta i l cen te r of pressure a t a = 160. corrected f o r h c h n u ~ b e r

Effect ive cen te r of prosaura of the incremental force on 0 body atrrkc-configuration due t o t he addi t ion of a t a i l

I n i t i a l s lope of t a i l chordwise cen te r of

p re r rure a t R - 160.

Strake leading edge s t a t i o n from doeet ip

Angle of a t t a ck ,

Angle a t which l i nea r var ia t ion of X begins CPB

Incremental CN on body alone 'due t o j e t - Cn 9P -%*

Incremental norm1 force coef f i c i an t due t o s t rakee

s lope of bcH vs a curve aAcN /aa 0s 0s '

XXV

11 1 B (ST) P

I n c r e m e n t ir, rrormiil f o r c e due t o t h e a d d i t i o n o f w i n g s t o a body

I n c r e w n t a l CN on s t r a k e s due t o j e t = C - C

N~~ N~

I n c r e m n t e l C on t a i l s due t o j e t = C - N N~~

Increment i n norn&l f o r c e due t o t h e t a i l s o f a ' b o d y - s t r a k e - t a i l c c m f i g u r a t i o n

T o t a l :ncremehta l CN on body + t a i l s c o n f i g u r a t i o n

due t o j e t e f f e c t s on t a i l s = (C + I B ( T ) p ) - (C + I B ( T ) )

*TP N~

I n c r r m e r r t a l normal f o r c e c o r i f i c l e n t p roduced on a t a i l due t o w i n g v o r t e x i n t e r f e r e n c e

I n c r e m n t a l i n t e r i e r e n a C c;n body due t o jet e f f e c t s on s t r a k e s a n d t a i % i = IB(ST)p =

I n c r r m n t a l i n t e r f e r e n c e C on b ' d y due t o J e t e f f e c t s on t a i l s a N- I

IB(ST)P B(T)

Spanwise d i s t ; i n c e ' be tween wing r o o t and l o c a t i o n , o f t r a i l i n g v o r t e x

Change i n CP l o c a t i o n o f s t r a k e + t a i l i n t e r f e r e n c e CN duc t o t h e j e t = S

C p ~ ~ - XCP,

Change i n CP l o c a t i o n o f s t rake-on-body i n t e r - f e r e n c e i n C due t o t h e j e t = - N X~~ 1 (S)P

Change i n CP l o c a t i o n o f s t r o k e + t a i l i u t e r - f e r e n c e 5 due t o jet e f f e c t s 011 XCp

S

Chn.ige i n CP l o c a t , i o n o f t h e s t r a k e + t a i l i ~ t e r - f e ~ e n c e 5 due to j e t e f f e c t s on X

, cp,

Change i n s t r a k e CP l o c a t i o n due t o l e t e f f e c t s - X c ~ -

S P XCP,

x x v i

Change i n t a i l 'CP location due t o j e t e f fec t s =

Difference between t a i l chordwise centers of pressure a t a -90.and any a < go.,

ae90

phi DifIsrence between t a q chordwise centers of pressure at a - 175. and 160'.

Mach number correction used a t a, = 160.

Change i n center of pressure

Vortex 'induced angle of at tack a t the t a i l

Crosaf$bw drag proportionality factor , 4

sweep angle

Taper r a t i o , t i p chord/root chord

Noadiarensionalized center of presmre = X /d CP

A f terbody

Basic

Body-rtrake

BST Body-~trake-tail

Body plus t a i l

BUdy In the presence of ' the t a i l

Body plue wing

B(W)

Bwr

C

D.P.

b e

I

i

L.E.

N

n

P

POT

ref

S

SF

S.P.

T

TO)

T.E.

v

W

W(B)

a

ABW

742

n

Body i n the prerence of a

Body p lur wing plus t a i l

Crorsf l o w

Double pure1

Expored

'B(T)

General indica tor

'Leading Edge

Nore

Nonlinear

Planform area

Potent ia l

Reference

Strake

Skin f r i c t i o n

Single panel

T a i l

Ta i l Ln presence of body

Tra i l ing edge

Vortex

Wing, o r wave dra8

'Wing i n the presence of a body

, Denote# d i f f e ren t i a t ion with rerpect t o a

SUBSCRIPTS (CONCL' D)

x x i x

Thin report d e ~ r i b e r the conr t ruct ien and ure of methods f o r pre-

d ic t ing the p i tch plane aerodynamic c h a r a c t e r i r t i c r of a c l a r r of m i m i l e

configurationr. The configuraticmr include body alone, body-tail,

body-rtrake-tail and body-wing-tail configurationr a t high angler of

at tack. An arreranent is a i r 0 provided of the e f f e c t r of a rocket exhaurt

flume on the p i tch plsne c h a r a c t e r i r t i c r for a range of th ru r t e r conditionr.

The u t h o d r , remi-ampirical i n nature, vere developed through corre-

l a t i o n of t a r t da ta obtained during w v e r a l independent t e r t programr.

There data, when taken together, form a ra ther extensive data bank i n

which configuration geometrier and flow conditionr a r e ryrtematically

varied. Except fo r the wthodr pertaining t o winged mi r r i l e configura-

tiorrr, which a re limited t o 30 degreer angle of a t tack, a11 method8 a r e

applicable t o anglem of a t t ack between 0 and 180 degree.. In severa l

inrtancer lack of t e r t da ta lopored M~ch number l imi ta t ionr ; hovever, i n

the majority of carer the method6 apply t o Mtch numberr between 0.6 and

3.0.

Methods a re provided t o predict the charac ta r i a t i c r of i ro la ted

components and interference e f f e c t r produced when variour component. a r e

combined. The method8 per ta in t o bodier of c i r cu la r crors-rection, When

t a i l s a r e added, they a r e mounted i n crucifona (plum a t t i t u d e ) with the

t a i l t r a i l i n g edger i n l i n e with the base of the body and undeflected.

Porvard l i f t i n g rurfaces ( r t raker or wings) can a l r o be aided.

The method. enable the urer t o ert imate the n o m l force and center

of prerrure of a var ie ty of configurationr by calculat ing the character-

i s t i c * of individual mis r f l e cooponentr and t h e i r mutual in teract ionr

XXX

produced when in combination. Uhtre poamible, predictiom have been

compared againat data which were not uscd In the develoglpant of modela.

In general, theme comp8rimona have demonatratad aood agreement.

xxxi

1 - 0 INTRODUCTICN

A r ecu r r ing problem i n m i s s i l e engineering is the lack of accura te methods

f o r p red i c t i ng configurat ion aerodynamic c h a r a t t e r i s t i c s , f o r a l l Mach numbers,

a t high angles of a t tack . The s i t u a t i o n is aggravated by the long term trend

toward increa ied m i s s i l e maneuverability and angle of attcick. H i s to r i ca l l y , '

, m a x i m u m ahgle requirements have increased s t ead i ly . The g r e a t e s t increase

has occurred r e l a t i v e l y recent ly t o meet advanced air-launched system

' maneuverability requirements. These now d i c t a t e angles of a t t a c k t o 90 and

, even 180 degrees.

The missiles which f l y a t these very high angles a r e usua l ly of the slewing

type, i .e . , t h e i r angle of a t t a c k is generated by t h r u s t vec tor con t ro l (TVC)

( f o r example, A I R SLEW and AGILE). Aerodynamically they tend t o be somewhat

simpler than mi s s i l e s which achieve high maneuverability through use of .

aerodynamic su r f ace de f l ec t i on because of t h e l a rge con t ro l fo rces ava i l ab l e

from the de f l ec t ed TVC nozzle. Non-TVC mis s i l e s w u a l l y can deploy wings and

canards ae wel l a s t a i l s , and t h e i r maximum angles of a t t a c k a r e l imi ted t o

about 40 degrees. A i r slew mis s i l e s usua l ly deploy t a i l s , bu t m y f o w a r d

l i f t i n g su r f aces a r e genera l ly small (e.g ., s t rakee) . Basic aerodynadc

predic t ion methods a r e required f o r both types of vehicles .

The aerodynamic performance of TVC type veh ic l e s ts f u r t h e r compli-

cated by plume in t e r f e r ence ; t he re fo re a method is required f o r calcu-

l a t i n g t h i s e f f e c t in addi t ion t o methods f o r es t imat ing the bas i c aero-

dynaad c s . It haa been well-establ ished (References 1, 2, 3, and 4) that t h e b e s t

means of constructing methods f o r emtimating bas ic aerodynamic character-

i s t i c a a t high angles of a t tack is through corre la t ion of experimental

data generated by t e s t ing over systematically-varied ranges of the relevant

geometric and a e r o d y n w c parameters (Reference 1) . This report describes

the generation of methods using t h a t technique. The methods deal with

the aerodynamics of aerodynamically controlled miss i les and TVC miss i les

with and without plume effects . A s u m a r j of the data -sad i n the develop-

ment of the methods is presented i n Reference 5.

The objective of t h i s work was t o evaluate exis t ing methods, t o

improve upon these exis t ing methods i f possible, and, where necessary, t o

develop new methods t o predict the pitch-plane aerodynamic charac te r i s t i c s

' f o r aerodynamically controlled and TVC missiles. The m~thods addressed

were applicable t o the configurations, angle of a t tack and Mach number

ranges indicated in Table I.

Table I

Scope of Methodology Requirements

CONFIGURATION

30dy Alone

1'

t Control Mechanism

Aerodynamically Control a - 0. - 30. M - 0.6 - 3.0

J

bdy-Wing-Tail (Canard)

bdy-Tail J

J

J J

TVC a - O* - 180' M - 0.6 .- 3.0

J

J

4 I

. J e t

Interference Effects Included

-

J

bdy-Strake

bdy-Strake-Tail

J

J

Prediction of the aerodynamic characeer i r t icr f o r the coafiguratioar indicated

in Table I requirer methodr f o r pradict ing ;he .erodyn.plicr of individual

components and mutual interference effect.. Figurer l a and l b rhow the

extent of ex i r t ing c a p a b i l i t i e r p r io r t o t h i r contract with rerpect t o t o t a l

methodology requiraamtr . ,

Although i t i r not r h m i n Figurer l a and b, a ce r t a in l e v e l of

capab i l i t i e r e x i r t d in each of the are- indicated. In general, the accuracy

of there methodr i r poor a t angler a rea te r than a few degreer; therr fore ,

there wthodr were not indicatud. Under the prerent work, methodology w u

developed t o f i l l In the gapo indicated i n the overa l l raquircnentr of

Figures 1. and b. The methodr developed a re of .n engineering type and

include char ts , graph6 ud formulation8 which f a c i l i t a t e e.se of w e by -hand.

By and large the methodr a r e empirical and therefore a re Limited t o the

: I range of t e s t condition8 and geometric parameters tested. The rpec i f i c

conditionr tes ted a re dircuared i n Section 2.0 and the Mach number range of / - . . i n t e r e s t , namely 0.6 t o 3.0 is adequately covered, However. a8 is usually the

care, the f l i g h t combination8 of Mach and Reynolds numbers were not achieved ,

in the wind tunnel t e a t p r o g r m . Therefore the resul t ing awthodr do not

contain a l l the e f f a c t r of Reynoldr nvmber var ia t ion that might be desired.

Unti l be t t e r matching of f l i g h t conditionr ir achieved i n vind tunnel t e r t r ,

the user of such wthodr muat exerc i re care and judgement with regard t o

. t h i r point .

izinally it ir noted t h a t methodology war developed t o predict induced

y w forcer and momentr and inducad r o l l i n g mmentr, and war provided u

par t of, t h i r program. Reference 39 describe8 the development of the methods

and th. conputarkad verrion of tha method@.

The general layout of the report ir u f o l l a w t P i r r t , a ~ a n e r a l

d e ~ c r i p t i o n of the equipment and models used i n data generation i r given i n

Section 2.0. Then a l imited amount of data analysis i r pramanted in Section

3.0. Pollowlng tNr, Saction 4.0 describer the formulation of the aerody-

namic prediction a q u t i o n r and the tetmr f o r which methodm a r e uonrtructed.

The methods themelves a re described i n Section 5.0. Where applicable each

description includes background dircuerionr, treatment of data, approach of

construction, use of methods, and where possible, check8 of method accuracy

against data not used i n the conetruction.

Existing

I_ JRequirementrr

b' -- 180

Angle of - 90 Attack ( d d ,

03.ch)

Figure la. Methodology Requirements for NC Mesilea

Existing Methodology

Attack

Figure lb. Methodology Requircacnte for Aerodynamically Controlled Xirriles

2.0 LXPBRMENTAL DATA SOURCES AND HDDELS

The major i ty of d a t a ava i l ab l e f o r c o r r e l a t i o n ( r e0 Figure 2) were

generated using e i t h e r U.S. Air Force o r Martin Mar ie t ta , Orlando Divi r ion ,

supplied models. Reference 13, which is based on 485 hours of t e s t i n g i n

tunnels 4T and A a t AEDC, is t h e primary source of da ta . The TVC d a t a a r e

taken from a 312 hbur t e s t program i n tunnels 16T and 16s a t AEDC. m i c e 1

m i s s i l e comporrents were teo ted sepa ra t e ly and in combination. A Martin

Mariet ta supplied r e f l ec tkon plane and f i n a were t e s t ed t o provide imolated

f i n da t a t o 160 degrees angle of a t tack . I so l a t ed body and non-rolled body

t a i l d a t a were generated using both A i r Force and Martin Mar ie t ta models.

The Martin Mariet ta main'body model is shown i n Figure 3a with the s e l e c t i o n

of t a i l s which can be mated t o t he body shown i n Figure 3b. The A i r Force

and Martin E l ~ r i e t t a models a r e both 10 c s l i b e r s i n length with tangent ogive

noses but the A i r Force nose is 2.5 ca l sbe r s compared t o 3.0 c a l i b e r s f o r

t he Hart in Mariet ta nose. The A i r Force and Martin Mariet ta model diameters

a r e 1 - 2 5 and 3.75 ' inches, respec t ive ly . T a i l s of i d e n t i c a l planform

geometry, arranged i n cruciform and undeflected, were t e s t ed on each body.

T a i l t aper r a t i o s , aspect r e t l o s and diameter t o span r a t i o e were var ied '

between 0 - 1.0, 0.5 - 2.0 and 0.3 - 0.5, respec t ive ly . Angles of

a t t a c k varied from 0 t o 180 degrees. The maximum angle of a t t a c k a t t a ined

by the Martin MarieLta stirrg mounted model was l i m i t e d , t o 60 degrees. Through

a combination o i s t i n g s and s t r u t s , t h e , A i r Force model was t ea t ed t o 180

degrees. The Martin Yar i e t t a model was equipped wi th four 3-component t a t 1

balances compared t o a s i n g l e t a i l balance f o r t h e A i r Force model. These

balances memured t'ail n o d force , hinge B a t and rco t bending moment.

Six-component main balance data were' available €ran tach model.

Body-wing-tail gonfiguratione were tested t o 30 degrees angle of a t tack

a t a non-rolled a l t i t u d e 'using the Martin Marietta model. Data c m r i r t a d

of 6-component nvin balance abd' 3-component f i n balance outputs. This

model can accampodate s e t s of half wings mounpd i n cruciform a t severa l

d i f f k e n l ax ia l s t a t ions between the shoulfizr and a f t e r body section containing '

the t a i l balances. The wings a re not attached t o recording balances. Wings

tes ted were of constant aspect r a t i o 2.0 and taper r a t i o 0.0 with diameter t o

span r a t i o varying between 0.35' and 0.5.

A pore complete description of the source8 of t e a t data, t e s t conditions

and model configurations is contained i n the Data Report (Reference 5) eubmitted

as part of t h i s study contract (CDRL Itm No. 'AOOS).

Figure 3a. Martin Marietta Main Body Model in the NSRDC 7'xlO' Transonic Tunnel a t S ixty Degrees kqgle of Attack

Figure 3b., Martin Marietta Tail b d e l s

3.0 AERODYNAMIC DATA TRENDS '

Befoze proceeding to the various methods, a qualitative analysis of

some of the teat data will be presented. The discussions are intended to

illuminate the basic phenomena underlying model aerodynamic behavior and

provide the user with more than simply a recipe for calculating the

vc.rious force and moment quantities. Many of the basic ideas used Mere

' presented in ~efcrences 2 and 3. They rill be sunmarired here for the

sake of convenience. The discuseions here will be limited to isolated

fins and bodies and body pluv tail configurations.

3.1 Pin Aerodynamics

Most of the discussions in this section are based upon those of

Reference 2. No attempt will be made to reproduce all of the previous

material. The reader is referred to the original document for a detailed

treatment.

The discuesions center on the effects of fin geometry (planform

taper and aspect ratios) and Mach number on the aerodynamic characteristics.

Pin f l m patterns are discussed briefly along with the aesociated stall

characteristics. The implications for fin normal force coefficient and

chordwise center of pressure location are outlined. Diecussions begin

with a consideration of delta fine.

' I

, Figure 4. Vortices Produced by the Reattachment of Lower Surface Boundary Layer

At high angle8 of attack the flow arouzd delta fins ir char-

acterized by the presence of large upper surface vorticea fed with vor-

ticity from the boundary layerr which reparate at the leading edges (See

Figure 4). Stall on auch wing8 in brought about by vortex "bursting".

Thi8 ir accompanied by 8 breakdown of the well-ordered vorttx flow and a

rudden prerrure increare at and downatream of th& "burrt" point. Upatream

the prerrure in the vortex remains lov and producer a ruction which in-

creaser the norul force. A. angle of attack ir increared tha '%urrtW

point rover up8ttuP touardo the tr8iling edge. When it crorrer the edge,

rtall baginr and is characterized by 8 lorr of n o w 1 force and a forvard

movement of the center of prerrure. An arpect rrtio increarer, the rtalling

angle crf attack decrearer. T b r e effect. are rhown in Figurer Sa and 5b

at tr~ronic speedr. The fi#urer alro rhow the follawing:

1 ) The no-1 force curve r loper, . a t a = 0' and, 180' "a

are nu&rically equal - t h i r remil t 1. predicted by

Slender Body Theory.

11) A t a = 90'. the centerr of prerrure and of area very

n u r l y coincide. Thir ir i n i t u i t h l y obviour.

111) A t a - 180°, the cu t t e r8 of prerrture of there d e l t a fin.

l ie rj*t a t the "leadily" edge. mi8 b& out the

Slander Body Theory r e r u l t tha t a11 of the loading 6n a ,

f i n occur8 over the region where the f i n rpan i r changing

( i n c r e u i l y ) . The predicted e f fec t of r e t rea t ing r ide

edaer (I...,, t o purh the center of prerrure u p r t r e u ) i e

not evldeht. A r i r i l a r r e r u l t i r found f o r non-delta

f in8 81.0.

S t i l l confining the d i rcurr ionr t o d e l t a fin., Figurer 5c and d

, ahow t h e i r behavior at ruperronic rpeeds. It w i l l be reen t h a t no r t a l -

1- i r v l r i b l e a t t h i r Mach number. During the re f l ec t ion plane t e r t r

from uhich there data were obtained, it war found tht near a = 90' a t

wperronic Uach nurberr, the f i n 8 behaved l i k e forward facing r t ep r , re-

a u l t i n s i n low valuer of $,. Accordingly. the CI( value a t a - 90' war

obtained from Reference 6 and the data fa i red through t h a t point a r rhown.

Alro uorthy of note i r the center of prersure behavior, pa r t i cu la r ly near

a = 180'.

When the f i n planform i r not t r i rngu l r r , the upper rurface vor t i ce r

referred t o e a r l i e r a r e m d i f i e d o r joined by yet other ro ta t ing flowr.

For rectangular fins,'the large suction-producing vortices now spring

from the side edges, whils a laminar separation bubble can exist at

the leading edge. When stall occurs on much a fin, it is frequently a

result of laminar bubble lengthening, spreading low-velocity, high

pressure flow over the upper surface. The result is a loss of normal

force and (I rearward shift of center of pressure. A clipped delta fin

'displays behavior somewhere between that of a delta and a rectangular

fin. This behavior is shown in Figures 6a and b at' transonic speeds.

Note the centers of pressure for the rectangle at a * 0' 'and 180'. They

lie right at the "leading" edge as predicted by Slender Body Theory. At

a - 180°, a11 three fins show this predicted behavior. As before. the

supersonic data show no visible stalling and have been faired through

from Reference 6, Figures 6c and d. c%2

The effect of increasing Mach number on a delta fin is to move the

vortex "burst" p~int downstream, Thus a fin which is stalled at one

Mach number may be unstalled by simply increasing Mach. This behavior is

shown in Figures 7a and b for an AR = 2.0 delta fin. The stalling

behavior at M - 0.8 is entirely removed at M - 1.3 and higher.

3.2 Body Aerodynamics

As in the case of fins, the aerody.1amic characteristics of

bodies at high angles of attack are largely influenced by viscous, sepa-

rated flows. The discussions belch deal with these, especially in the

case where the body wake takes the form of an asyuunetric vortex pattern.

This phenomenon has recently become of considerable interast for high

incidence miss i les (Reference 7).

When a slender miss i le body is placed a t angle of a t tack in . . uni-

form flow, ' the boundary layer generally separates on e i t h e r s ide of the

body and £oms a lee-side wake. Separation usually begins near the rea r

when the miss i le reaches about 6 degrees angle of at tack. The woke takes

the form of a pa i r of,aymmetrically-dispored, counter-rotating vor t ices

fed by vor t i c r ty shed from the separating toundary layer. Ae angle of

, at tack increases, the u t i a l extents, s i zes and strengths of vor t ices

increase also.

When the body angle of a t tack reaches about 25 degrees, the ryamct-

t i c a l pature of the wake disappears. The two vor t ices are joined by a

th i rd , beginning again a t the body rea r , and the wake becomes aryrmnetric. Ae

angle of a t tack i m increased fur ther , more vor t ices jo in the flow u n t i l the

wake contains several which have been ahed from the body. A section

taken through the wake shows i t s t o resemble the von Karman vortex s t r e e t ,

v e l l &own in the l i t e r a t u r e on two-dimensional f lows.

The asymmetric nature of the wake produces an as~?naetric d i s t r ibu t ion

of pressure forces along the body. This r e s u l t s i n out-of-plane forces

and moments being induced, whether the body has l i f t i n g surfaces deployed or

not. These forces and mments can be s ign i f i can t ly large, requiring specia l

means t o be found t o counteract or remoye t h e i r a k f e c t s (Reference 8) .

Figure 8a shave the force and leoment coeff ic ients induced La a body a t

M I - 0.6. The e f fec t of increasing Mach number t o supersonic values is

uwuaily t o reduce those ef fect6 t o negligible prc;ort,ions. This nmy

be seen i n Figure 8b for M - 2.0. Later discusstons w i l l i l l u s t r a t e the

d d i t i m a l e f f e c t s of adding l i f t i n g ru r f ace r t o such a body. The ateady,

asyumetric wake p e r s i s t e up t o angles of about SO t o 60 degrees. A t higher

angles the wake becmea unrteady m d v o r t i c e r are ahad &8ymmetrically.

3.3 Body T a i l Configuration Aerodynamicc

The addi t ion of t a i l s t o a body genera l ly increaaer the out-of-

plane forces and mocPents induced by a rymwtr i c vortex e f f e c t s a s we l l aa

producing r o l l i n g moments. Several axampler w i l l be g i w n of these

Important e f f e c t s . Figures 9a and b r h w out-of-plane q u a n t i t i e s a t M - 0.6

f o r two t y p i c a l sets of cruciform tailr f ixed t o t h e 10:l c a l i b e r body

("glw"' a t t i t u d e ) . It ir of i n t e r e a t t o note the correrpondence between the

peaks of force and mament. The ang le of a t t a c k has genera l ly been l i m i t i d :.

t o 90 degrees because:

I ) By 90 degrees the wake flaw is unsteady and the out-of-plane .

q u h t i t i e r f l u c t u a t e rapidly. .

11) Above 90 degreer , the preaence of the s t r u t rupport might ca&

a l t e r a t i o n s i n the wake pa i t e rn and its e f f e c t a .

By the tim. Mach ,number haa reached 2.0, no induced e f f e c t 8 .re v i r i b l e . .

(not shown here). ,

Another i l l umt ra t ion of t he asymmetric wake e f f e c t is contained i n

Figpres 10a.and b. Revioua t e r t i n g on a PP(O model with four i na t ruwnted

tails yielded the forcer and moments on t he indiv idual t a i l r . Complete

c o n f i g u r a t i n r o l l i n g . w e n t was obtained from separa te (main balance)

instrunwntation. Figure 10. rhows the t a i l forces f o r a "crosr" configurat ion

(+ - 45 ' ) a t angles of a t t a c k t o 60 degrrea. I f t he mmentr of t he re t a i l

R O O Q

0 0 0

0 b i ,

0 0 91 4 B .G

YAWING NOHENT COEFFICIENT AND SLDE FORCE COEFFICIENT

P l u s T o i l

............ Rolling Maarent

I

1.1 pigure 9.. O y t a f - P l m e Forces And M o t a m t ~

'$0 V o r t e x AsFQetn

p l u s T a i l , dB - 0.5 , A = 1.0, - 0.5

-- 1 I I

0 = 45 D e g r e e s I I

A W L E OF ATTACK-DEC.

F i g u r e fO.. Comparison O f T a i l N o r m a l F o r c e s

Figure lob. ~op~htison Of Rolling Homents

-- ANGLE OR ATTACK-DEC.

4.0 PORMlfATION OF THE A E R O D W C PREDICTSOU EQUATIONS

brccrusa of the nature of the information available, the following

formulatione of bodytai l , body-mtrakttall and body-wing-tail configuratfou

pitch-plane 8erodynaanCc charrcteri~tice are aaceoaary. Theee formulations

vill vary depeadirg on vhether the configurations are to be aerodynamScally

or thruat vector controlled (TVC).

Aerodpnamically Controlled

body- ~811

' , Body-Strake-Tail

Thrust Vector Controlled

Body-Tail,

C + 2 $ %(B) 2 'c'T (B) + d S~ d

'CPTP + (TI , d , d.'

Body-Strake-Tail

Hence, the following quantities are required in order t o conduct

aerodynamic analyees on body-tall, body-wing-tail, or bedy-strake-tail

configurations vhich are either aerodynamically or thrust vector controlled.

The aection of this report in which each quantfcy is developed is listed

. ' Quantity

xcp, (T)

Section Page

Quantity Section P.ga

5.5.1 310

*CP, 5.5.2 323

AC %P 5.5.3 334

A. indicated above, certain of the quantities are applicable to the

equations for aerodynamic control as well as the equations for NC. Others

are used only in the TVC model. L i m i t s of applicability for each method arc

indicated in the appropriate sections.

5.0 AERODYNAMIC METHODS -

5.1 Isolated Components

5.1.1 Body Normal Force

A method is presented f o r predict ing body normal force coef f i c i en t s ,

CN , fo r angles of a t tack between 0 and 180 degrees and Mach numbers B

from 0.6 up t o 3.0. Comparisons between predicted r e s u l t s and experi-

mental data.ahow good agreement. This'method represents an improvement

over exia t ing methods i n tha t it accurately predic ts CN both transonically B

and supersonically.

hckground

The aerodynamic force directed normal t o a body i n its pi tch plane

can be separated i n t o potent ia l and viscoue flow contributions. Using

slender body theory, Munk found the potent ia l flow contribution t o be

equal t o s i n 2a, wherz 2 is the slope of the normal force coeff ic ient

curve a t a - 0 degrees. In l a t e r work by Ward (Reference 9 ) , i t was shown

tha t t h i s force is actual ly directed midway between the normal t o the

stream and the normal t o the body axis . Taking t h i s in to account, poteri-

t i a l contributions t o body normal force can be expressed as :

' a - #in 2a coe T

A t very

force.

e f f e c t s

low angles of: a t tack; t h i s potent ia l term domi'nates body 17ormal

However, fo r angles of a t tack greater than 6 degreee, viscous

a r e introduced and rapidly become the dominating factor. Existing

theor ies do not adequately predict viscous e f fec t s . Empirical procedures

have bee5 developed based on the e a r l y work by Allen and perkinslo and

~ e l l y ~ ' which introduced the concept tha t the viscous crossf low around

incl ined 'bodies of revolution is analogous t o the flow around a c i rcu la r

cylinder normal t o the flow. i n accordance with standard notation, these

empirical procedures r e l a t e the viscous normal force contribution t o Cd , C

the crossflow drag coeff ic ient deftned by analogy v i t h twodimensional

flow. Thus

~xper imenta l data have rhown Cd t o be a functlon of both Reynolds and C

crorsflow Mach numbers. Values of , I have been determined empiricallv

from tmdimeneional and f i n i t e length cylinder data. , ,

Combining the theore t i ca l potent ia l and empirical viecous contr i -

bution r e s u l t s i n the following expression f o r body t o t a l normal force

coef f i c i e n t :

This ii the ur exprersion uaed by .Jorgensenl* t o predict t r a n s o n t and

auperwnic value. of CN for ' angles of a t t a c k between 0 and 180 degrees.

The procedure outlined by Jorgensen i n Reference 12 was found t o be

inaccurate a t transonic Mach numberr when predicted reeu l t r were compared

wi:.~. th &ta of Reference 13. There camparimnr a r e prersnted i n

Figurer 11 through 14. Accuracy i r only f a i r when a11 Mach numberr and

-lea of a t t ack a r e conridered, but doer improve with increaming Pkch

number.

P.m avenue. a r e avai lable t o improve accuracy. l i r r t , develop a new

method t o improve traneonic capab i l i t i e r . The mcond, and p e r b p s moat

desirable approach, would be t o develop a s i ng l e procedure which would be

accura te both t r ad ron i ca l l y and supersonically.

Method Develovmeat

A power s e r i e s approach is used t o develop a method which pred ic t s

t he combination of po t en t i a l and viscous e f f e c t e on body t o t a l C N'

Boundary condit ibns were sought vhic h would adequately def ine the character-

i s t i c s of CN between angles of a t t a ck of 0 and 180 degrees. Values of ,

CN and - a t a = 0 n/l,and n vere take. a s bpundary condit ions. aa

Experimental da t a indicated t ha t h u e s of C a t a = 0 and a = n a r e N

zero. Also from experimental da ta , i t was observed t ha t - - 0 a t a - r/2 aa

a = ~ .nd re The r m i n i n g boundary condi t ioar , 1.0.. CN a t a * ~ / 2 and - at a - 0, aa

were re ta ined a. f r e e variables.

Applying these boundary condit ions t o t he expreaaion:

2 3 4 - a + a a + & a + a a + a 4 a + a 5 a 5 54, 0 1 2 3

yielded

which can be rewr i t ten a s

Values of A and A a r e p l o t t e d 5 a s func t ions of angle of a t t a c k i n 1 2 ,

Figure8 15 and 16. Values of CN and CN still r equ i r e d e f i n i t i o n . 4 2 a

Transonic va lues of C presented i n Figure 17 a s a func t ion of

' Mach number, nose length and af terbody length were taken from Reference

, . 14 and 15. Supersonic va lues of CN presented i n Figure 18 were taken a

from Reference 16 a e a funct ion of Mach number, nose l eng th and af terbody

length. The data . of Figures 17 and 18 represent improvements over

e x i s t i n g co r r e l a t i one . Linear i l t e r p o l a t i o n i e required f o r values 'of

CN between Mach 1.2 and 1.5. a

Values of CN can be ca lcu la ted wi th itquaelon 1 3 recognizing n12

t h a t t he "potent ial" term goes t o zero and u t i l i z i n g the published d a t a

f o r va lues of n (Reference 17) and Cdc(Reference, 12). The a v a i l a b l e

valuesl of n (shown as no I n Figure 19; a r e derived from subsonic t e s t

da t a and a r e t y p i c a l l y assumed t o apply up t o c rass f low MPchnumber (n c )

equal 1.0. Above Mach one n is normally asuumed t o be 1.0. Rather than

continue t o use such a discontinuous representa t ion ,a procedure is

employed here which produces an eet imate of t he v a r i a t i o n of n with Mc

through the t ransonic regime. The t r a n s a n i c , v a r i a t i o n of a is developed

a s follows:

The p o t e n t i a l component of normal force is still defined a s i n

Equation 11 with t h e change t h a t CN rep laces t he 2. The i n t e n t is t o a

make uee of t he t e s t da t a (Reference 13) a s a oource f o r CN r a t h e r than a

r e l y on the t heo re t i ca l va lue of 2. Then the viscous cont r ibut ion t o

t h e normal fo rce is defined a s a s f o ' l l a a :

kS = 5 - a in ($( a) cos (a/2) a

5 and 5 are both obtained from t h e t a e t da ta . Then a G.

n c = 9 1 s Dc 2 + e i n a

3 r e f

The quan t i t y 1 t+, ins ca l cu l a t ed , u t i l i z i n g t h i s expreesion a t crossf low C

Mach numbere ranging from 0.2 t o 2.0. Values af C were taken ftom *c

Reference 20 a t the corresponding Mach numbers t o permit so lv ing f o r n.

The curve f a i r e d through t h e va lues of TI h i c h r e s u l t from t h i a exerc iae

, is shown i n Figure 20. The subsonic va lue i e seen t o apply up t o about

Mc C.8 with t h e ,upward ,trend continuing t o about Mc, = 1.4. ' A

polynomial expression was then derived a s followe t o represent t h e

v a r i a t i o n of 0 v i t h Mc,

- as = 0.0 a t F$ - 0.8 and 1.4

n = no a t Mc = 0.8

n = 1.0 a t Mc = 1.4

Applying these boundary condi t ions t o t he fol lowing expansion:

?.

TI = aO + alMc + a;W,: - - a3MC 3

yielded

s - no (-9.0741 + 31.1111 Mc -30.5556 42 + 9,2593 3) C

3 + (10.0741 - 31.1111 MC + 30.5556 nC2 - 9.2593 Mc )

which can be r ewr i t t en as:

rl = Bo Oo + Bl

Equation 16 i a applicable t o crossflow Mach numberr between 0.8 and 1.4

Values of no a r e contained i n Figure 19. Values of B and B1 a r e pre- 0

sented In Figure 21,

Values of Cd from Reference 13 modified on the baei r of the C

r e s u l t s of Refereme 3 a r e presented i n Figure 22. There data cover a

wide range of crossflow Mach numbers and c o w from a number of d i f fe ren t

sources.

Using the above information and Equation 12, it is now possible t o

ca lcu la te the value of C required f o r the c a l c u l a t i m of CN between Nm/2

o - 0 and 180 degrees.

K e t b ~ d Evaluation .-

Check cases were made using the same configuration and conditions

represented i n Figures 11 through 14. Figures 23 th roNh 26 rhow com-

parisons between these predict ions, experimental data, and predict ions

using Jorgensen's procedure (Reference 12). These comparisons indicate

improved accuracy a t high angles of a t t ack i n the transonic Mach regime

and equally good accuracy a t a l l angleo of a t t ack i n the supersonic

regime.

Use of Method

The method f o r predict ing i so la ted body normal force is applied i n

the following way.

1 Depending upon the Mach number, use either Figure 17 or 18 to - determine CN as a function of nose and afterbody length.

a 2 Calculate the value of CN using Equation 12. -

r/2 a Use Figure 22 to determine C . -

dc b Depending upon the Mach number, determine the value of n. -

. For Mc 5 0.8, use Figure 19 to determine rl as a function

of t/d.

. For 0.8 < Mc * 1.4, use Eqwtion 16 and Figure 19.

For Mc * 1.4, = 1.0.

3 Using Equation 14, the results of eteps 1 and 2,and Figures 15 - and 16, calculate the values of C from 0 to 180 degrees.

N~ Numerical Example

Calculate C~~between 0, and 180 degrees at M = 2.86 for a body with

the following characteristics:

% - - 3.0 (tangent ogive) d

2 Uae the followinp equation to calculate CN - n/2

c ~ d 2 .4

* Sret

a From Figure 22, Cd - 1.34S~C

b For M - 2.86, n = 1.0

c Therefore CNI2 - 13.67

3 Using the following equation and Figures 15 and 16, calculate

CKB iACNO + A2 CNw2 SrefinS.ase

"A, A22;

0 1.0 0.0 0.05 0.074 0.01 0.36

10 0.123 0.045 0.98915 0.153 0.095 1.7620 0.167 0.155 2.6330 0.162 0.305 4.6640 0.13 0.475 6.8950 0.09 0.645 9.0960 0.051 0.79 10.9570 0.023 0.905 12.4480 0.005 0.975 13,3485 0.001 0.99 13.5490 0.0 1.0 13.6795 0.001 0.99 13.54

100 0.004 0.975 13.34110 0.015 0.905 12.42120 0.026 0.79 10.88130 0.034 0.645 8.92140 0.037 0.475 6.61150 0.033 0.305 4.27160 0.022 0.155 2.19165 0.014 0.095 1.34170 0.007 0.045 0.636175 0.002 0.01 0.143180 0.0 0.0 0.0

Data Comparisons

The results of the numerical example are compared with experimental

data,(Reference 18) in Figure 27. Because these data were not involved In the

development of the method, this comparison represents an independent

check of the method. Agreement is quite good throughout the angle

46

of attack range tranaonically. F i p r e 28 represeat. further ilulepeadent

checks of predicred resu l t@ .painat experimental data from Reference 19.

Coaparihns between predicted reeul tu and experitnental data have shorn

the method of t h i s eection t o be more accurate than the Jorgenaen method

i n the W l o r i t y of cases. However. the Jorgenaen method has proven more

accurate i n the 0 t o 40 degree angle of a t t ack range t r anson icd ly .

Therefore. it is ~ecgpm)gnded that the Jorgensen method be w e d in t h i s

region and the method of t h i s sec t ion h a11 others.

Figure 11. Camparison of hperbental and Predicted Results (C ) , Mach - 0.6 N~

0 20, 40 (Q W 100 120 140 160 180

UELI O? *RACK - DlcuLI

Figure 12. Comparison of Experimental and Predicted Results (CN ) , Mach - 1.1) B ,

0 40 60 80 100 120 140 I60 180 A W L I 01 ATTACK. DUImS

Figure 13. Comparison of Experimental and Predicted Results (C ) , Phch - 1.30 N~

0 10 40 (0 0 1 0 120 140 1W 180 U l E L l W ATTACK. DmIW

Figure 14. Caparison of ~ x ~ e r i r e n t a i and P t d i c t e d Results (C ) , Mach 1 2.0 *B

Flsure 15. Coeffleiente for Calculation of C

4

Figure 16. Coefficients for Celculatlon of C %

Figure 17a. Curves for Transonic '. ('tq/d = 1.5) %

Figure 17b. Curves for Transonic C , (11~/d - 2.5) Na

a Figure 17c. Curves for Transonic CN ( N/d = 3 . 5 )

a

Figure 18a. Curves for Supersonic C ' (tN/d 'm 2 -51 Na

, ,

Figure 18b. Curves for Supersonic C (tN/d - 3 .0 ) Na

5 3

Figure. l ac . Curves f o r Supersonic CN ( s / d = 3.5) a

Figure 13d. Curves f o r Supersonic CN (tN/d = 4.0) a

Figure 19. Correlation Factor for End Effects

Figure '20. Variation of n With Mach Number

C R Q S S V M MACH warn, n,

Figure 21. Curves fo r Determining Traneonic Valuae of n

Figure 22b. Croesflow Drag Coefficient (Subcritical Crossflow, Mc < 0.4) ,

Figure 23. Comparison of Experimental and Predicted Results ( ) , Wch 0.6 "a

mL1 OI ATTACK, DDCULS

Figure 24. Cornperison of Experimental and Predicted Results <C ) . Mach - 1.15 Na

% ' .

- - -- ~ Y P sn1s.s

0 2J 40 W 80 LOO 120 140 160 180

UCLll 01 AlTACK. DffiUXS

Figure 25. Conparidon of Experimental and Predicted Results (C ) , Mach - 1.30 N~

0 20 40 w 80 100 120 140 160 UCLI (X A R A C I . DD3EICT I@?

Figure 26. Comparison of Experimental and Predicted Results (C ) , b c h - 2.0 *B

MCLI O? ATTACK. DffiRMS

Figure 27. Comparison of Experimental and Predicted Results (C ) , ~ a c h N~

Figure 28. Comparison of Experimental and Predicted Results (C 1, Mach - 0.85, 1.20, and 2.25 N~

5.1.2 Body Center of Pressure

Summary

A method is presented f o r p r ed i c t i ng i s o l a t e d body cen t e r of p r e s su re ,

XCPg, f o r ang l e s of a t t a c k between 0 and 180 degrees and Mach numbers

from 0.6 up t o 3.0. Comparisons between pred ic ted r e s u l t s and experimental

d a t a show good agreement.

Background

New highly maneuverable missiles w l l l encounter extreme anglee of

a t t a c k . I n some ca se s angles of a t t a c k may approach 180 degrees i n

e i t h e r t h e t ransonic o r supersonic Mach regimes.

E f f ec t i ve eva lua t ion of proposed conf igura t ions w i l l r equ i r e methods

f o r p r ed i c t i ng aerodynamic c h a r a c t e r i s t i c s a t extreme ang l e s of a t t a c k

over a wide range of Mach numbers. Current p r ed i c t i ve techniques a r e

l im i t ed t o angles of a t t a c k l e s s than 30 degrees. New methods a r e required

t o f i l l t h e void between e x i s t i n g and required c a p a b i l i t i e s . This s ec t i on

d e a l s s p e c i f i c a l l y with a method for ' p r ed i c t i ng body cen t e r of p ressure ,

tPg. The method presented is app l i cab l e t o Mach numbers between 0.6

and 3.0 and angles of a t t a c k between 0 and 180 degrees.

Method Development

The method f o r p r ed i c t i ng XCp was developed using an empir ica l

approach. The i n i t i a l s t e p involved a survey of ava i l ab l e d a t a (References

, 13, 18, and 19 ) . The d a t a displayed c h a r a c t e r i s t i c s which were unique

t o s p e c i f i c Mach number and angle of a t t a c k ranges. For Mach numbers of

1.0 o r g r e a t e r , XCP displayed a rapid rearward movement between angles

of a t t a c k of 0 and 20 degrees, followed by a nea r ly l i n e a r progression

o f XCP between 20 and 160 degrees and passes through the cen t ro id of t he

planform a rea a t 90 degree. F ina l l y , betveen 160 and, 180 degrees'. another

6 1

rap id rearward movement of XCp v a s observed. Experimental d a t a shomd

t h a t t h e XCp l e f t t h e body b e t w e n 170 and 180 degrees. A s tln body

approacher 180 degrees , a cpuple is produced a s the p o s i t i v e p o t e n t i a l

normal f m e on t h e Corvlrrd fac ing p o r t i o n of t h e body become8 equa l t o

the nega t ive p o t e n t i a l f o r c e on tha t r a i l i n g nose por t ion of the body. This

couple s u b j e c t s t h e body t o moment and t o a zera n e t normal force . Under

t h e r e circumstances, c a l c u l a t e d va lues of XCp tend to become i n f i n i t e l y

l a rge .

€or llech numbers l8se than 1.0, XCp d i sp layed t h e same c h a r a c t e r i s t i c s

between 0 and 20, degrees and 160 and 180 degrees. However, t h e l o c a t i o n

of X tended C'o remain e s s e n t i a l l y cons tan t between 20 and 50 degrees , CP,

followed by a rearward movement v t ~ i c h is l i n e a r between 50 and 160 degrees

and passes through t h e c e n t r o i d of t h e planform area a t 90 degrees .

A power s e r i e s approach was used t o develop t h e method between 0 '

and 20 degrees. I n the usua l way boundary cond i t ions were sought. The

c q n t e r of p ressure a t a - 0 degrees was taken as t h e f i r s t boundary

condi t ion. Curves p resen t ing X as c func t ion of lN, lA, aad E; i n +lo -- d d -

t h e t r anson ic Mach regime a r e presented i n Reference 3. For t h e sake

of completeness t h e s e a r e presented aga in he re i n Figure 29. S imi la r

d a t a i n t h e supersonic Hach regime (1.5 2 H 2 4 . 5 ) were found i n

Reference 16 and a r e presented i n Figure 30. For a second boundary

cond i t ion i t can be s h m t h a t f o r s y m e t r i c a l bodies 3 ~ ~ ~ / ~ l - - 0.0. aa , o

A t h i r d boundary cond i t ion was def ined by t h e c e n t e r of p ressure a t 20

degrees. Thfs was de f ined a c the c e n t e r o f pressure a t zero d e ~ r e e s

,p lus an i n c r m n t . Using data f r o r References 3, 13, and 20, t h e

percentage of body length by which XCp s h i f t e d between 0 and 20 degrees was

detarmdned a e a func t ion of Mach number (see Figure 31). As a f i n a l boundary

condi t ion , a t 20 degrees was assumed t o equal t he d o p e of t h e aa

l i n e a r v a r i a t i o n between 20 and 90 degrees ang le of a t t ack . Experimental

d a t a Indicated t h a t t he cen t e r of pressure a t 90 degrees could he approximeted

a s t he cent ro id of t he planfonn area . A t 90 degrees, when the flow is

separated along the e n t i r e length of t h e body, t he normal fo rce w i l l be

due e n t i r e l y t o crossf low drag (Reference 3). Assuming a cons tan t .'d along C

the body, t he c e n t e r s of pressure and of planform a r e a should then coincide.

Col lec t ing boundary condi t ions and applying them t o the following

polynomial expansion

yielded

X - which can be r ewr i t t en a s

Where

Values of Ao, A and A2 a r e p lot ted a s a function of angle of a t t ack i n 1

Figure 32.

Equation 17 was developed based on the charac te r i s t i c s of XCp a t Mach

nimbers of 1.0 or greater . Applying Equation 17 fo r Mach numbers l e s s than I

1.0 w i l l produce good r e s u l t s even though aX a t am20 degrees w i l l be CP - in error. aa

A s indicated e a r l i e r , the va r i a t ion i n XCp between 20 and 160 degrees

is dependent upon Mach number. For Mach numbers l e s s than 1.0, the

location of X remains conetant between 20 and 50 degrees and then moves CP

l i n e a r l y toward the rea r t o t h t value of X a t 160 degrees, paesing through CP

the centroid of the planform area a t 90 degrees. For Mach numbers of 1.0

o r greater , XCp va r i e s l i n e a r l y between the locations a t 20 and 160

degrees, passing thoough the centroid of the planform area a t 90 degrees.

Using t h i s information, the following equations were derived f o r determining

the slope of the l i n e a r va r i a t ion and the value of x a t 160 degrees.

(19)

where a * , the angle mrk ing the bound of the low angle region, is 20 degree8

fo r Mach numbers of 1.0 o r q r u t e t and 50 degrees f o r Mach n ~ ~ b e r s :$..re than

1.0.

A power s e r i e s apprarch #a used t o develop the wthod betueen ZW .nd

180 degrees and i n the u.ur1 wry boundary coadltionr uerr sought. The renter

of pressure at 160 degrees w s trhr a s t b first boudaq c.adft fee-

, This can be calculated using Equation 19. A second boundary. $ ( m i t t e n xa)

at 160 degrees was assumed t o equal the e lo re of the l i n e a r va r i a t ion between

a' and 160 degzees, This value can be calculated using Equation 18. Also,

a s a th i rd Wundary condition it can be shown tha t a t 180 degrees aX CP 10.

aa A s a f i n a l boundary condition, the center of pressure a t 180 degrees was

assumed equal t o the body length, r a the r than trying to define i t a s some

point off the body a s indicated e a r l i e r . This assumption w i l l intrcduce

no , s ignif icant e r r o r s sinc,e the resu l t ing forces and,moments a r e m a l l .

Collecting these boundary conditions and applying them t o the

following polynomial expansion

yielding

X = -51840000 + 900000 a -5200 a2 + 10 a r 4000 'a1 60

Use of Metnod

Tire method f o r predictlag i so la ted body center of pressure is applied

as follows:

Depending upon the Mach regime, uee either Figures 29 o r 30 to

detenj .ne xo as a function of tN/d and gA/d. Linearly in terpola te

fo r v a h e a of xo betveen Mach 1.2 and 1.5.

U ~ l l n g Figure 31. determine the rearward s h i f t i n center of pressure

between 0 and 20 degree. for the appropriate L/d and MH. Add this

value t o the resu l t 'of Step 1 t o determine Q ~ . Calculate the distance from the nose t o the' centroid of the planf orm

area using

.. , and where S and S a r e t h e p l a n f o n areas of the nose and cyl indr ica l

PN PA sections respectively i n the ease of a tangentagive cylinder body

S 2 -1 e~ P, =, &N JW + +R s i n R - -2(R-r)

and

Note tha t - */2 *

d

4 Using Equation (17). t he r e r u l t s of s t e p s 1, 2, and 3, and Figure 32, - c a l c u l a t e t he c e n t e r s of p ressure bctweqn 0 and 20 degrees.

5 Calcula te t he s l ope of x a t 150 degreee using Equation (:a). - 6 Calcula te t he va lue of x a t 160 degrees using Equation (19) - I Using Equation @ ) , t h e r e s u l t s of Steps 5 and 6, and Figure 33, -

c a l c u l a t e t he c e n t e r s of preseure !wtween 160 and 180 degreee.

8 Depending upon the Mach n w b e r rt.nGe of i n t e r e s t , determine - t he variation of x between .?O and ~ O ! J degrees.

a. For M 1.0, ektend a s t r a i g h t l i n e from xZO t o x ~ ~ ~ .

b. For H < 1.0, maintain 4 cons tan t value of x from 20 t o

50 degrees and then extend a s t r a i g h t l i n e between

t he va luer of x a t 50 degrees and 160 degreer.

Numerical Example

Calcu lc te x between 0 and 180 degrees a t M * 2.86 f o r a body with t h e

following c h a r a c t e r i s t i c s :

3.0 tangent - ogive

6.0

g/d = 9.0

d - 1.5 inches

1 In t e rpo l a t i ng between t he va lue r ofFigure 30band30c, x0 war - ca lcu la ted t o be 1.93 c a l i b e r s a f t of t h e noee.

2 Uring Figure 31. AX/L/d = 0.285 a t M = 2.86. Therefore, f o r t / d = 9, - AX = 2.565.

Uee the following equation ard F i ~ u r e 32to calculate the centers

of prrseurc between '0 and 20 degrees.

5 Using the following equation, calculate the d o p e of the linear - variation betveen " Q and 160 degrec~ .

6 Using the followfng equation, calculate the value of x at 160 degrees. -

7 Using the following equctton and Figure 33,calrulate the,centcre of - pressure between 160 m d 180 degrees.

8 Graphically determine values of x between 20 and 160 degrees by - connecting x and x with a straight line.

20 160

Data Comparisons

he result a of the nimerical example are compared' against experimental

data from Reference 18 in Figure 34. Because the data were not involved i n

the development of the methods, this cornpariaon represents an independent

chCck of the methods. Agreement is good throughout the angle of attack

range. .Figures 35, 36, 37 and 38 prenent further comparisons with other

experimental data (~eferences 13 and 19). Again agreement is quite good

in all cases, except for the higher angles of attack in Figure 36.

Figure 29a. Transonic Tangent Ogive-Cylinder Zero Angle of Attack Centers of Pressure

(LN/d - 3.5)

Figure 30. Tangent Ogive-Cylinder Zero Ar~gle o f Attack Centers of Pressure

0 I I I MI '9mlUI

Fi8ure 31. Increment i r k Center of Preaourc Betveen Angles of Attack of 0 and 20 Degrees

0..

"b I . .

Figure 32. Polynouinal Coefficienra, L w A n ~ l e of Attack

ANGLE O f AlTACX % DECKCICS

Figure 33. Polynominal Coefficients, High Angle of Attack

Figure 34. Comparisons Between Predictions and Experimental Data XCpdd. Xach = 2.86 ,

Figure 35. Comparisons Between Predictions and Experimental Data XCpB/d. Mach 2.25

Figure 3 6 . Comparison Retween Predictions and Experimental Cata

Figure 37. Comparfsona Betveen Pred ic t ions and Experimental Data XcYB/d, W., i - 0.80

Figure 38. Comparisons Between Predictions and Expe-imental Data

5.1.3 Body Axia l Force

Summary

Methods a r e p resen ted f o r p r e d i c t i n g CAB t h e i d o l a t e d body a x i a l f o r c e

c o e f f i c i e n t . Angle of a t t a c k and Mach ranges a r e 0 - 180 degrees and 0.6 - 3.0, r e e p e c t i v % l y . hro methods a r e used. I n t h e s u p e r r o n i c range a modif i -

c a t i o n of a n e x i s t i n g technique due t o Jorgeneen (Reference 1 2 ) is recommended;

i n t h e t r a n s o n i c range, a nev method based on an e x t e n s i o n of a p rev ious

t echn ique has been c o n s t r u c t e d . The o v e r a l l performance of t h e methhds is

ehovn t o be good.

Background

An examinat ion of e x i s t i n g methods f o r c a l c u l a t i n g body a x i a l f o r c e

c o e f f i c i e n t from 0 - 180 degrees J i s c l o s e d t h e fol lowing:

1 The method o f Jorgensen (Reference 121, which a p p l i e s o v e r t h e e n t i r e - a n g l e of a t t a c k range, is a p p l ? c a l l c - n l y t o supt-rsonic k c h numbers.

2 The method of S a f f e l l , Howerd and Brooks (Reference 21). which uses - almost t h e same formula t ions a s Jorgensen, d e a l s v i t h l i f t a~id drag ,

r a t h e r than normal and ax ia l f o r c e components.

3 The wehod of F i d l e r and Bateman (Reference 3) . which a p p l i e s over - t h e a n g l e c f a t t a c k range 0 - 90 degrees , i e s p p i i c a b l e t o

t r a n s o n i c speeds .

Because of i t s inconvenience, p l u s i ts s t r o n g s i m i l a r i t y t o t h e

Jargeneen method, t h e work of Reference 21 v a s not , conaidered f a r t h e r .

I n s t e a d , References 3 and 12 were examined t o d e t t m l - s v h e t h e r t h e formcr

needed t o be improved f o r superson ic speeds and t h c l a t t e r could be modified

t o app ly from 0 - 180 degrees f o r t r a n s o n i c sperds . The s u p e r s o n i c and

t t a n s o n i c ranges a r e d i s c u s s e d s e p a r a t e l y . CA is taken ~ o s i t l v e when

d i r e c t e d towards t h e base .

Method Development (Transonic Mach Nunbere)

The basic method here is tha t of Reference 3 which appl ies from 0 GO

90 degrees. T h bur ic formulation of the predict ion equation is:

where f(n,o) - f(M.90) - 0, and

C, - C5 + ,C*base (CAI includes wave and f r i c t i a n e f f e c t s 0

C h r r t s w i l l be presented for estfmating a l l t he quan t i t i e s required.

. For 90 - 189 degrees, the following formulations were devised empirical ly

L a . , by choosing functional forms which a r e consistent with th; pa t te rns '

observed i n the test data.

CA - C, - (CA - C, ) s i n a* , 90" - < a 0 0 lf

Tbe base drag contr ibution is obtained from Section 4.2.3.1 of Ref. 17.

CA - CA for blunt cylindere and is obtained from Reference 6, from

1

which Figure 39 is reproduced.

Use of the bas ic and modified formulations of Reference 3 provides the

estimetee which a r e compared with data i n Figures 40a - f . It w i l l be seen

thac nvltchlng is qu i t e good ove ra l l and t h i s method is recommended f o r use.

Use of Method (Transonic Mach Numbers)

Restating the basic equations:

is obtained from Figures 41a - c a6 follows:

1 Prom Figures 41a, b, 'and $ determine CA , the basic axial lb

6 force coeff ieient (excluding base drag) at Reb - 15.8 x 10 .

2 From Figure 42 determine the scaling factor Q1/CA1 at the b

b required R then CA = C A1 e' 1 A1

2 From Section 4.2.3.1 of Reference 17, find C %ase

I , f (M,a) is a power series containing a and the value of CA at a - A'

I 70". as a free variable. The power series is:

I For conlenience, Figures 43 and 44 are given which are sufficient co calculate

f ( H , R ) . Figure 44 presents f (M,o) for various kigure 4 3 presents A '

values for various transonic Mach numbers. Use of the curves is as

4 At the appropriate M, read C - from Figure 43 - A

5 In Figure 44 estimate f(M,a) over che angle range at - the appropriate

A'

Numerical Example

Calculate the axial force coefficient -.ariation between 0 and 180 degrees

for a missile body having a 3 caliber'tangent ogive nose followed by a 7

caliber cylindrical section. Machnumber is 1.15. G ' 3 - 8 (10'~)

1 From Figure 41 (int~rcnlating) CA = 0.215. - lb

2 From Section 4.2.3.1' of Reference 17, C - 0.05. %re

Hence, CA + C o - Aba8e

I The variation in thu axial for& coefficient with angle of attack ir

detarmlned ar follow:

0 - 90' 4 - Now ki - 1.15, hence - -0.52 (Figure 43)

f (l4.a) (Fin. 44)

0

-0.11

-0.26

CA - C,, + f 04.a)

0.455

0.455 Sac Figure 41.

A - 0.455

0.156

-0.324

-0.666

-1.162

-1.34 See Fiuure 414.

160 - 180' CA- C% -1.34 (Figure 39)

Method of Development (Supersonic Mach Numberel

In Reference 12, equation8 and charte are presented from which the

axial force coefficj-nt on an isolated body may be eetimated. The equations

ueed are

2 CA = CA cog a - c a 2 90' 0

The zero-angle coefficient is expressed as:

where C A ~ , ChSF and C A ~ ~ ~ ~ are the contributionc due to forebody preeaure,

skin friction and baee drag, respectively, while the 180 degree coefficient *

(flat baee into flow) ie given by:

Charts a r e presented f o r estimating a l l ' t h e above quant'5ties except

"SF . For purposes o i the 'preeent work, the skin f r i c t i o n was estimated

from Saction 4.1.5.1 of keference '17, assuming a turbulent boundary layer.

Predictions of CA over the e n t i r e angle of a t t a c k r a n g e were made' using

the Jorgensen formulations. Comparieone between predict ions and data a r e

show?! acrose the eupereonic Mach number range i n Figures 45a - d , It should

be notad tha t the AeDC wind tunnel data (Reference 13) a r e uncorrec&d f o r

e r e pressure ef fects . It w i l l be seen tha t matching is reasonably good,

' but obvious discrepancies a r e evident. For example, from 0 - 90 degreee

the data do not approach zero a8 predicted. They e i t h e r remain f a i r l y

constant o r CA increaeee e l ight ly . Also, rrom 90 - 180 degrees some die-

crepancies a r e observable.

It was decided t h a t a simple modification t o the method could eas i ly

be accomplished t o improve its performance. Based on the formulations

which were found to work fo r the transonic case, the followfng empirical

equations a r e used.

C A , - CA - (CA - CA ) s i n a' 90. 2 a 2 160' 0 0 1

a. - f (a - 90)

CA is s t i l l obtained from the techniques of Reference 12. 0

The remults of applying these equations a r e shown i n Figures 45a - , d .

Clearly the overa l l matching is b e t t k than before. The modified equations

a re recommended for use instead of Reference 12.

Numerical Exam - l a

Eatimate the axial force coeffictent variation from 0 - 180 degrees for a

miae.ile body having a 3 : l caliber tangent ogive nose v i th a 7: l caliber

cylindrical afterbody. Mach number is 2,O.

+ C Now - + doe

0

- 0.357 (Reference 12)

CA - -1.66 (Reference 12) ' U

0.357 See Figure 45b -0.092

. . A I 1 1 . 1 1

1.0 1.4 1.8 2.2 2.6 3.0 MACH NUMBER

Figure 39. Variatio~ With Mach Number of 180-Degree Axial Force Coefficient (Reference 6) ,

ANG1.E OF ATTAqK - DEC.

ANC1.E OF ATTACK - DEG.

- ,PREDTCTTON 0 ' DATA (REF. 1 3 )

(c) M = 0 . 9 --

Figure 40. Comparison Between Predicted and Rxperimental C (TrnnsnnIc) A 8

I ANGLE OF ATTACK - DEG, h

(d) M = 1.0

120 140 160 iao MGLE OF ATTACK - DEG.

- , , PREDICTION

0 DATA (REF. 13)

Figure 40, Continued

Figure 41. Curves for Determining CA

b

F i g u ~ z 41. Continued ,

Figure 42. Scaling Factor for CA

b

- Fibure 43. Variation O f CA ~ i t h ' h a c h Number

AWCLK OF A R M X % DECREES

Figure 44. Baaic Curvea of f(M, a) Calculated From Power Seriec

Flgure 45. Comparisons 0e:vcen Prediction and Experimental CA* (Supwsonlc)

5.1.4 Pin Normal Force

The normal force coefficient of an isola ted t a i l panel C , can be pte- *T

dieted by the empitical methoda dc'. .loped iri Reference 2 and extended by the

correlat ion8 presented i n th in rectioq. The applicable range of the methods

is now:

h c h number - 0.60 t o 3.0

Ampact r a t i o - - c 2l.0

Taper r a t i o - 0 t o 1.0

Comparisons between predict ions and t e a t data ahow generally good agree-

ment, normal force coeff ic ient being predicted usually w e l l within 10 percent.

Thin wthod allows the normal force coeff ic ient of low aspect r a t i o f i n s

t o W calculated fo r angles of a t t ack from 0 t o 180 degrees and fo r Mach

ncrmbers ranging from 0.6 t o 3.0. The method is an extension of the method

presented i n Reference 2, Section 3.3.1, made possible by the acquis i t ion of

addit ional t e s t data. Typically, the method consis ts 'of two operations:

1) a procedure t o eetimate CNT up t o 30 degrees, and 2) a procedure t o extend

the estimate t o 90 degrees. It is shown tha t a mirror image of the curve so

obtained provides a good estimate t o 180 degrees. A t supersonic Mach numbers

greater than about 2.5, 8 eingle procedure is shown t o f i t the t e s t data

adequately within PO percent.

Method Ikvelopmcnt - A f u l l d w e l o p e n t cf t h i s method is contained i n Reference 2. A portion

of t M t material w i l l be included here f o r completeness.

The f l a , about low aepeFt r a t i o f i n s i t snglca of a t t r . g r r t e r than

A few degrees is characterized by non-linear phenomena that :annot be described

by linear theory techniques. Techniques have been dev-'.pel f o r predict ion

of the aerodynamic charac te r i s t i c s of low aapect r8;i.J f ino i n the presence

of upper surface vort ices, e.g., Referencee 22, 23, 24, 25. However, f o r

the general case where vort ices, both coherent and burs t , and a laminar

separation bubble a r e present (Reference 26). methods a r e not available.

The method developed here is derived from the popular crossflow drag

based methods a s typi f ied by Reference 23 which eaploys the formulation:

% = C l a + C 2 a 2 (21)

In t h i s expression C 1 is the zero angle of a t t ack normal force curve slope and

C2 i e a constant chosen t o force the expressisn t o f i t experimental data.

This equation may be regarded a s a truncated power s e r i e s i n a. Since

It contains two const&s, it should f i t tw boundary conditions on CN, the

condition, (C ) = 0 having already been s a t i s f ied. One condition is Nogo

that (a cN/ea)a-O must equal the normal force curve slope a t a-0. Tiis I

1 determines C1. The second :ondition, the one determining the value of C2, is

I usually chosen so t h a t the experimental data a r e f i t t e d a t some high angle of

at tack. The expression is then reasonablp accurate up t o tha t angle of a t t a c k ,

provided that the curvature of the data carve always has the same sign. In the

general case, an expression such a s Equation (21) cannot adequately describe

the shape of the normal force curve above a few degrees angle of at tack.

Since t h i s form of solution leaves many boundary conditions unsatisfied,

an obvious means of improving t h i s s i tua t ion is t o r e t a i n the power se r i e s

form of expansion, but t o include a s many terms a s the boundary condicions

Six boundary conditionr a r e iden t i f i ed for ume i n a more general expreaaion along with a comwnt a r t o the b a r i r f o r the choren value.

CNT(0) - 0, theoret ica l and empirical

cy,dO) theoret ica l

CNT (n/2) from t e a t data

The quanti ty CR(0) is s e t t o zero on the bas i s of l inea r theory predic-

t lons a s well a s t e a t data whereas the valuee of the o ther f ive quan t i t i e r a r e

h u e d on a review of t e s t data. Am indicated below, the resu l t ing power s e r i e s

is expressed i n t e r n of the tw non-zero boundary conditions, I..,, Cw (0) a

Power Ser ies Solution

With the s j x boundary conditions av.ilable, a power se r fes containing

I mix unknown coef f i c i en t s m y be uaed. The power s e r i e s i m asaumed t o be of

the form: 5 , c ~ ~ ( a ) - A,, a*

0

I from which, with the a id of the boundary conditions, the s i x unknown constants

Ag through A5 m y be determined. Substi tut ion of boundary conditions and

rearrangement of the equation yields:

clir(a) - CHI (01 a + (n12) L%dQ- I - a

(I I

where a Is i n radians.

T5 f a c i l i t a t e computation of the power r e r i e r , Equation (23) has been

ru r ranged i n the form:

C b ( a ) ' A(a)CNT (0) + B(a)CNT(n/2) a (24)

where the values of A(a) a d B(a) a r a function of angle o f , a t t a c k a r e shown

i n F i p r e 46.

The term C N ~ (0) murt be obtained from l i n e a r theory o r Reference 27 a

from which Figurer 47a through 47d a r e taken . The C N ~ ( W / ~ ) term is obtained

through comparison of C b ( 4 with experimental data.

The quanti ty cNT (n/2) i r empirically derived and presented i n Figure 48.

It i r important t o note tha t the numerical values assigned t o the boundary

condition CNT(n/2) a r e not ac tua l normal force data but ra ther expediently

choren numbers which produced good agreement i n the angle of a t t ack range

between 0 and 30 degrees. Even so, t h i s approach does not work uniformly

f o r a l l geometric8 and Mach numbers and yet another device i e required t o

complete the port ion of the model up t o 30 degrees. Toward t h i s end a

quanti ty a' is defined which marks the upper bound angle of a t t ack t o which

Equation 24 applies. I f the value of a ' found i n Figure 49 is l e s s than 30

degrees, then the value of C N ~ f o r a between a ' and 30 degrees i r obtained

by adding an incrament (ACN) t o the value of C N ~ obtained a t a'. That i n ,

f o r anglee of a t t ack between a ' and 30 degrees, l e t :

where: ACN - ( $ ! ) A C H ~

Th. quanti ty A C ~ I A C ~ is an empirically determined factor (Figure 50) ranging

from 0 t o 1 indicating the f rac t ion of the a m x i m u m correction (ACm) required.

The quanti ty ACm i r an empirically determined parameter (Figure 51) reprmenting

the laager t d i f ference found betwmen the C& calculated a t any a' and the t e s t

value8 a t a - 30 degreer.

Uas of Method

The normal force coeff ic ient mathod conr i s t s of Equation (23) o r (24),

together with values of cNT(r/2) from Figure 48, rupplemnted with addi t ional

infomat ion from Figurer 49 through 51 where required. The following r t eps

I wrt be adhered to.

1 Calculate CNT (0) uaing Figure6 47. - 47d (or wA~f2'for lawtrt. AR'r) - a

I 2 Find C N ~ (n/2) from Figure 48, intmrpolating where necessary - I 3 Calculate CN~(O) up to a' a r obtained from Figure 49. . (If a' -

30 degrees, calculat ions a r e now complete. If a' 30 degrees, go

I on t o Step 4).

I 4 For Mach numbers under consideration, obtain values of AC~/ACN a t - M

, v&riour (a - a' ) / ( 3 0 - ' a') from Figure 5C.

5 From Figure 51 find use t h i s t o ca lcula te valuee of ACN and - I d i r t r i b u t e theee over the a range from a' to 30 degrees.

Numerical Example

'hro axamplea i l l u s t r a t e application of the method.

Calculate the var ia t ion of normal force coeff ic ient with angle of a t tack

t o 30 degrees f o r a wing a s follows:

AR - 0.52, A ' - 0 , M - 0.8

1 CHT,(O) - nAR/2 - 0.819 from Slender Body Theory

2 CNT(n/2) - 2.4 (Figure 48)

3 Figure 49, a' = 30 degrees, hence c8lcul8te f (a) up t o t h i s value. -

Uring Equation (23). the following table m y be conrttucted:

5 0.091 A comparioon betveen there valuer and experi- 10 0.216 mental data taken from Reference 24 i r shown 15 0.368 i n Figure 52. Alro o h m 10 the resul t of 20 0.541 applying the mathod of Reference 17. The 25 0.727 prewnt method yieldr bet ter matching with

' 30 0.927 data.

Calculate the variation of normal force coefficient vi th angle of a t tack

to 30 degreer for a vfng am follow8 :

AR - 2.0, A - 1.0, I4 - 0.98

(PigUte 47.)

(Figure 48)

12 degreer, which i r < 30 degrees; hence, CN is calculated

ouly up t o a - 12 degrees:

4 Figurer 50 and 51 m e t be used fo r a > 12 degreea. Therefore, from Figure 50, e

a t l b l e of ACN/CH *(a-a* ) / (30-0' ) is constructed: M

calculated. I

The moat d i rec t way t o obtain the norm1 curve t t o draw CNT up t o a'

and then add, on the remining CN increments. This curve i c compared with

experimental ,data i n Figure 53.

One fu r the r check case is shown in' ~ i g u r e 54 f o r a f i n of AR - 0.86,

X - 0.4, and li - 1.02. b a i n data mutching is qu i t e good. These data were

obtained from Reference 28.

Extension of Method t o 90 Degrees

I n extending the methnd to 90 degrees angle of a t tack, a second power

, s e r i e s i e ueed along with data (Referencee 5 and 29) a t angles between 30

degreee and 90 degrees. Due t o the lack of deta i led high angle data a t

low supersonic apeede, i t ha8 not been poesible t o check the method a t a l l '

Mach numbers. However, check cas te using the high angle traneonic data

reforred t o e a r l i e r have ehown the method t o y ie ld reasonable accuracy.

The procedure ueed t o extend the method t o 90 degreee introducee a

se r i e s (similar t o Equation (23)) which joins the curve fo r C N ~ a t 30 degrees

2 t o the point a t 90 degreee (CN . Experimental values of C N , ~ from transonic

t e s t s on the f i n e deecribed i n Reference 2, abd ~upereonic values from

lhference 6 combine t o produce the c u m I n Figure 55.

Boundary condi t ionr ured i n conr t ruc t ion of the new s e r i e s are:

1 A t a - 3 0 degree., C ~ ~ ( 3 0 ) may be determined from the f i r a t p a r t of - thlr method.

2 A t a - 30 degree., C b m y be da temined from the d i f f e r e n t i a l form - of Equation (23).

with a - 0.523 rado (30 degrees) f o r M > 1.

For Steps 1 and 2, however, va luer of CNT (30) and C (30) have been N ~ a

obtained from the experimental d a t a and are given i n Figures 56 and 57,

rarpect ively.

3 A t a - 90 degreer, CN i r determined from experimental da t a a s shown - C i n Figure 55.

4 A t a - 90 degrees, aCN/a a - 0 - \

Uslnq the re boundary condi t ionr i n t he a e r i e r of Equation (22) yields:

C b ( a J - 1.178111 CNT 0 0 ) + Ce(n/2) - Cm(30) + ~ ~ ( 3 0 ) ) a

3.7501 Cba(30) - 4.29731

a + - 2.342356 cNT (30) + 5

+ '1 0.911921 C* a (30) - (25)

I t rhould be m t e d t h a t Equation (25) ha8 been reevaluated and the

conrtants a r e rouewhat d i f fe ren t from those i n Reference 2.

For u r e i n ca lcula t ing CN*(Q) from 30-90 dagreer, the form of Equaticn

(25) ha8 been rearranged cu followr:

- C(a) %(3O) + D(a) C N ~ + E (a) (30) (26

with the threa t a m , C(a), D(a), and E(a) rhom i n Flgure 58 a s a function

of a-le of attack.

,U.a of Method (30' 1 a d90.1

Betmen 30 and 90 degrees 8-1. of a t tack, the method i8 used a s follows:

1 Find %(30) from Pigure 56 - 1 Find C h (30) from Figure 57 - a

3 Determine CN from F u u r e 55 - C 4 Calculate Cm(a) ualng Equation (25) o r (26) -

It i r raconmnnded tha t C N ~ be calculated from Q u a t i m (25) o r (26) beginning

a t a - 50 degrees and tha t the port ionr of the curve from 0 - 30 dagreea and

from 50-90 degraer be fa i red together. A 8 an example, the var ia t ion from

0 t o 90 degree. a-le of a t t ack i r c ~ l c u l a t e d f o r a f i n having aspect r a t i o

1.0 and taper r a t i o 1.0 a t a M~ch number of 1.10:

c ~ c - 1.42 (Figure 55)

C N ~ (30) - 1.98 (Figure 57) a

Ch(30) - 1.23 (Figure 56)

Subr t i tu t ing the above valuer i n Equation (26) yieldr:

Cw - 1.23 C(a) + 1.43 D(a) + 1.98 B(a)

Thir q u a t i o n i r u u d t o conrtruct the following table:

The cospariron between there emtinut- and the high a q l e data (Reference

5) ir rhovn i n Figure 59, While, exact matchi* is not achieved, the curve

does follow the data qu i t e well.

The method a s darcribed has been found t o require a minor modification

f o r f i a r of taper r a t i o 0 and aspect r a t i o 1.0 when Mach number i r l e e r than

1.0. When ertimating the normal' force coeff ic ient of an i so la ted f i n having

t h i s geametry a t H z 1.0 the following modification t o the method is suggested.

' i ) Use the f i r s t par t of the prathod, a8 described, ,up t o a - 30 defraes

i i ) Inrtead of uring thc recond power r e r i e s given by Equation (25) o r

Equation (26) fo r f a i r ing from 30 - 90 degrerr follow . the r t eps

below:

I r t imate A C N ~ f o r angles bet,wen 30 and 40 degrees from

Figure 60 and add t o the value a t a - 30, from (i) above.

h i r the curve from the value a t 43 d s g r ~ e s t o the 90

degree value, C N ~ from Figure 55.

E?EE&

l r t ima te values of C N ~ between 30 and 40 degrees fo r an isola ted f i n of

taper r a t i o OB aspect r a t i o 1.0 a t M = 0.9.

From Figure 60B the portion AB of the curve is used exactly a s shown,

i.e., add the increment taken from AB d i r e c t l y t o the value calcuAated a t

a - 30 degrees. The port icn BC of ?St curve applies between a - 35 and 40

degreea and must be scaled by the fac tor 0.78 (Figure 60b). Between a - 35

' I

and 4 0 degree* the l n c r m n t is equal t o t he product of the m a l e f ac to r and

Ax. The r a ru l t i ng c u m i r rhovn i n Figure 61. Point "A" muat now be placed

a t tb value of C b a t 30 degree8 f r o l (i) abop., and the c u m from point "C"

mrt be f a i r ed t o C w a t 90 degrees.

E x t a r i o n of Method t o 180 Degrees Angle of Attack ,

The a v a i l a b i l i t y of addi t iona l t a r t data on low aspect r a t i o t a i l panels . t o 180 degree. angle of a t t ack ha9 permitted the predic t ibn wchods t o be

extended t o t h i s nev range. The v a l i d i t y of the method i n the t ransonic arld

low ruperronic rpeed range is fu r the r demonstrated i n Figure 62 and Figure 63.

I n there f igures , the predict ion method employs the power s e r i e s of Equation

(24) up t o 30 degrees angle of a t t ack , and Equation (26) from 50 t o 90 degrees,

with the curve f e l r ed from 30 t o 50 degrees a s described e a r l i e r . Note t ha t

t a a t data from 90 t o 180 degrees a r e p lo t t ed on these f igures , indicat ing near

e r n t r y of t he data about 90 degrees angle of at tack. The predic t ion is

show t o c lose ly approximate t e s t da t a over the angle of a t t ack range from

0 t o 180 degrees.

A typ ica l appl tca t ion of t he method a t subsonic speeds, M - 0.6, and

from 0 t o 180 degrrca anale of a t t ack is shown i n Figure 66 ind ica t ing the

predrct ion nethod used f o r the vsr ious segments of the curve. The data a r e

again seen t o be nearly symmetrical a b w t 90 degrees and c lose ly approximated

by the predict ion mtthods.

Extension of Predict ion Methods t o H - 3.0

Supersonic t e a t data t o Mach 3.0 and 180 degrees angle of a t t ack f o r the

family of low a s p e ~ t r a t i o , l ov t r p e r r a t i o pa r~e l s were examined f o r canpati-

b i l i t y with the predict ion methods used f o r the rubsonic and transonic caees.

It rhould be noted here tha t durin, supersonic t e s t s some of the i so la ted

panels on the r e f l ec t i on plane encountered flow separation when the f i n s were

m a r l y ~ O N I t o the flow. The f i n r behaved l i k e f o m r d f a c i q r tepe, with

the ruult tha t CN war reduced. Here the CN value a t 90 degreer wr obtained

from the f l a t p l a t e data of Figure 55 and the t e a t data were fa i red through

tha apparent reparation rrgion.

8xaairution of the developed amthodr indicate tha t a8 the curve f i t

parameter Cb(w/2) of Figure 48 approximate8 the ac tua l value of normal force

at 90 degreer, CN , of Figure 55, and e ingle power ae r i e s method of Equation C

(24) ehould adequately predic t the va r i a t ion of CN with alpha. Typical ' T

examples w i l l be presented a t Mach 1.0, 2.5, and 3.0 t o i l l u s t r a t e the capa-

b i l i t i e s of the two povltr r e r i e r approach, and t o rhow tha t the s ing le power

a e r i e r of Equation (24) i r a teasoruble predict ion method a s the Mach number

approaches the 2.5 t o 3.0 range. The mirror image charac te r i e t i c of the

superwnic data about 90 d e g ~ e e r i r a l s o evident from the t e s t data, permitting

the predict ion t o be applied t o 180 degreer a t these Mach numbere.

A typical configuration example w i l l be examined a t h c h 2.0 where the

value of Q+("/2) i r acmewhat l a rge r than CN . In t h i r case, the two power C

ee r i e r approach yield8 a very good match with t e r t data an seen i n Figure 65.

Bad the method of Equtfon (24) been extended above 30 degreer the equation w u l d

h v e predicted a value of CNT("/2) vhich a t tNr Mach nurnber would be somewhat

hiah. Thir a d othar t e a t case8 ind ica te tha t a t Mach 2.0 the predict ion

rhould include both power r e r i a r , Equation (24) and Equation (26), i n order

t o obtain the beat data f i t .

A t b c h 2.5 tha r e q u i r e n u t f o r uring the two eegment predict ion method

begin8 t o disappear a8 tha valuer of cW(n/2) and , C N ~ r t a r t t o converge. A

typical example of a r ing le curve f i t from Equation (24) i r ahown i n Figure 66,

a d ir reen t o reasonably patch the t e s t data. The dashed l i n e of Figure 66

ahow that the curve f i t can be improved s l i g h t l y by the applicat ion of the

recond power re r iee , Equation (26). from 50-90 degree8 angle of at tack. I n

general, the a iag le curve ftt , Equation (24), rhould begin t o be acceptable

a t t h i r Mach number and above,

' h e appl ica t ion of the s ing l e power ro r i ea , Equation 14, predict ion , ,

method a t Mach 3.0 ir rhom i n Figure 67 and Flgure 68 f o r tm typ ica l t a i l

panel conf igurat ions, ind ica t ing good ag reemnt v i t h test data.

Figure 46. Power Scriao parameta& for Equation '(24)

Figure 47c. Lift Curve Slope for Taper Ratio - 0.25

MACH NUMBER

UACH UWRER

Figure 48. Variation- of CN ( n f 2 ) With Mach Number T

Figure 49. a'. Angle of Attaek ~bovt'which ACI( Hurt Be Applied (Subronic Only)

1 .0

MACH

0.9

0.8

Figure 50. Dimeneionleee CN Increment Above a'

i 0 .8 end 0 .9

TAPER RATIO, A

Figure 51. ACyl Hatimum Increment of Normal Force Above a' '

(Subronic Only)

Figure 52. Conrpariron of Predicted and Experimental C N ~ , Mach - 0.8

r e 3 . Comparison of Predicted and Experimental C N ~ . Mnch - 0.98 '

Flgure 54. Capariaon of Predicted kid Experimental C N ~ , Mach = 1.02

Figure 56. Variation of Normal Force Coefficient C (JO), With Mach Number, a - 30 Degrees N~

MACH NUMBER

MACH NUMBER

MACH NUMBER

Figure 57. Variation of CN (30) With Mach Number

*a

ANGLE OF ATTACK s DEGREES

- - ANGLE OF ATTACK DEGREES

Figure 58. Power Series Parameters for Equation (26)

AMGLE OF AlfACK % DECREES KACH M M I E R

Pigura 59. Comparison of Predic.ted and Experimental CN* Prom 30 to 90 Degreer ,

32 36 40 0 AMGLE OF AlfACK % DECREES KACH M M I E R

Figure 60. Curves for Modifying CN Method (A-0, AR-1.0, Subsonic)

C *T

PLOTTED VS. (180-a)

0 20 40 60 SO 100

ANGLE OF ATTACK, DECREES

o :I n 40 60 80 1 no ANGLE OF ATTACK, DECREES

'pigure 62. Comparison of Method and T e s t , C;gT at H = 1.0 and 1.3

(A - 0, AR = 0.5)

.. , 0 20 40 60 SO 100

ANGLE OF ATTACK. DEGREES'

0 20 60 60 80 100

ANGLE OP ATTACK, DEGREES

pigure 63. Comparison of Method and Test, CNT at r M - 1.0 (A = 0.5, AR - 0.5) and

H = 1.3 (A = 0.0, AR = 1.0)

9 ' 20 40 A0 100 189 60 , 140 160 1

U C L E OF ATTACK. DECREES

ANGLE OF ATTACK, DECREES

Figure 66. Compariaon of Test and Method, M - 2.5 (CN~)

Figure 65. Campariaon of Tert,and Method, Mach - 2.0 (CN~)

ANGLE OF ATTACK. DECREES

Figure 67. Comparison of Test and Method, Mach = 3.0 (CNT), A = 1.0, AR 1.0

ANGLE OF ATTACK. DECREES

Figure 68. Comparison of Test 'and Method, Mach * 3.0 (CN~), X = 0.0, AR - 1.0,

5.1.5 Fin Chordwise Center of Pressure

Surm~ary

A method is presented t o predic t XC , the chordwlee center of p~

pramsure f o r low aspect r a t i o f ina. The method i r va l id f o r angle8 of

&:tack up t o 180 degrees, and f o r Mach numbers i n the range of 0.60 t o 3.0.

The method is an extension of the method presented i n Reference 2, Section

3.3.2, 'and was made possible by the addi t ional t e s t data of Reference 13.

The corre la t ion method is shown t o predict s a t i s f a c t o r i l y center of pressure

location on typical miss i le f ins . The r e s u l t s of t h i s btudy apply t o

i sola ted l i f t i n g surfaces a s well a s t o undeflected wings o r tails fixed

t o missi le bodies. The l a t t e r asser t ion is based on comparisons presented

i n ~ e f e r e n c e 2. The method is divided in to two main divisions: 1 ) A

procedure f o r estimating chordwise center of pressure a t angles of a t t ack

to 90 degrees, and 2) A procedure fo r extending the est imates fo r angles

between 90 and 180 degrees. ,

Background

The development of the f i r s t pa r t (a - 0 t o 90 degrees) of the method

is contained in ~ e f e r e n c e 2. A portion of tha t material w i l l be included

here f o r completeness.

Three basic theorles: 1 ) Slender body theory, 2) S t r i p theory, and

3) Linear ( f i n alone) theory, a r e currently used i n predict ing chordwise

center of pressure. These theories have been found to provide f a i r r e s u l t s

a t low angles of a t tack. However, a s angle of a t t ack is increased beyond

the l inea r l i f t curve slope region the r e s u l t s become erroneous. Slender

body and s t r i p theory have been combined i n developing a method f o r

predicting chordwise cel.:er of pressure of a f i n tha t is attached t o a

cy l indr ica l body. For a t r iangular f i n , Reference 30 shows tha t a l l three

methodm give essen t i a l ly the same r e s u l t s f o r the chordwise center of

pressure of a f i p in the presence of the body. This might have been ex-

pected, mince the presence of a body induces an upwash t h a t changes f i n

loading in the spanwise di rec t ion, thus having l i t t l e e f fec t on the chord-

wise load dis t r ibut ion. Reference 30 a l s o showed tha t the f i n alone l i n e a r

theory is bes t f o r representing the chordwise center of pressure of low

a8pect r a t i o f ins . Xowever, due t o the i n a b i l i t y of t h i s theory t o predict

accurately f i n center of pressure beyond the region of l inea r l i f t , addi t ional

predict ion methodology i n t h i s area obviouely is needed.

Method Development

To generate the methods, center of pressure chordwise location f o r a ,

t a i l alone was calculated from normal force and hinge moment ttst data

(References 13 and 31). These t e s t s featured isola ted t a i l panels mounted

on re f l ec t ion planes that were deflected (rotated) throughout a range of

a - 0 t o 180 degrees. The Mach number range of these t e s t ,data is from

Mach 0.60 t o 3.0. Ta i l panel geometric parameters include three aspect

r a t i o s (AR = 0.5, 1.0 and 2.0) and three taper r a t i o s (A = 0.0, 0.5 and

1.0). The chordwise center of pressure locat ion is referenced t o the juncture

of the t a i l panel leadiag edge and re f l ec t ion plane, and the resul t ing center

of pressure is non-djmensionalized on the bas is of panel root chord,

The data were analyzed fo r e i m i l a r i t i e s and s ign i f i can t para-

meters, knowing t h a t the expression f o r the location of the center of

pressure is, i n general,

I txuination of the data rhowed tha t AR war the l e a r t r i g n i f i c m t of the

above pa rawte r r . This impliar t h a t the depmdency of hinge Promant on AR

is due to the dependency of normal force on t h i s quantdty. Keeping i n mind

X ' tha t ' AR is not a strong parameter, the expresrion f o r 2 i r defined a s a

C~ function of a and A a t selected value8 of A . and Mach 0.98.

The dercr ip t ion of the method proceeds a s follows. Presented f i r s t is

the technique used t o cor re la te the var ia t ion i n center of preseure posit ion

with angle of a t t ack a t a fixed Mach number (basic Mach 0.98) f o r various

combinationa of aspect r a t i o and taper r a t io . It is noted tha t the va r i a t ion

with a is subdivided in to four subsets coneirt ing of ; a = go0, 0 $ a < go0,

Following t t e corre la t ion a t M = 0.98, a technique i e presented which

X permits ca1cu:ation of the 3 fo r Mach numbers between 0.6 and 3.0.

c~ Region I (a - 90 d e ~ r e e e )

The chordwiee center of pressure of a t a i l panel a t 90 degrees can be

thought of cs the focal point or bas is f o r the cor re la t i cn method. The

aerodynamic 'loading of (i t a i i panel positioned normal t o the flow (a = 90 v

"CP degrees) is considered to be uniform. To the extent tha t t h i s is t rue , -

C~ of the tail w i l l coincide with the centroid of the panel area. In

addit?.on, t h i s re lv t lonshlp should be independent of Mach number. References

13 and 31 show t h i s asseswnent t o be v d i d . The t e s t data f o r a A. - 0.5

t a i l panel can be seen i n Figure 69. A s rhown, the area centroid of the

panel and center of pressure nearly coincide a t a = 90 degrees, Therefors,

a t a = 90 degrees the chordwise center of prereure (XCp/CR) w i l l be

determfned by the area centroid. For a nonscuept t a i l i n g edge plenform,

area centroid is only a function taper r a t i o a s rhom i n Figure 70.

It should be noted here tha t boundary layer separation occurred on

the re f l ec t ion plane %n f ron t of the r a i l s during some of the supersonic

t e s t s when the f i n s were nearly normal to the flow. The f i n s behaved l i k e

forward facing s teps , with the r e s u l t t h a t the cN data is rendered meaning- T

l e s s and eo was discarded. To f i l l the gapsthe data from the subsonic

t e s t s , where separation did not occur, were supplemented with supersonic

data from Reference 6 on f l a t p la tes normal t o the stream i n place of the

supersonic t e s t data where separation occurred. The l a t t e r data proved t o

be compatible with the data generated during Martin Marietta t e s t programs.

Region I1 (a < 90 degrees)

Examination of the a = 0 t o 180 degree data , as shown i n Figure 69,

shows a , smooth va r i a t ion i n ' c e n t e r of pressure i n the v i c i n i t y of a = 90

degrees. The method used f o r predict ing XCp/CR a t angles of a t t ack below

90 degrees was presented i n Reference 2. This procedure has been extended

i n Mach number and w i l l be res ta ted f o r completeness.

The re la t ionship between XCp/CR and a is plotted i n Figure 71

f o r the three t e s t aspect r a t i o s (AR = 0.5, 1.0 and 2.0) a t the basic Mach

number 0.98. Mach number e f f e c t s on XCp/CR must be included i n such a

manner tha t w i l l allow its influence t o vanish a t a = 90 degrees. Thus, f o r

the region a < 90 degrees, the chordwise center of pressure i e given by:

where - -

Therefore, a t any angle of a t t ack Peas than 90 degrees, the d i f ference

i n XCp/CR batmen the pa r t i cu la r a i n question and tha t a t 90 degrees, is

subtracted from the 90 degree (centroLd of area) value. For Mach 0.98,

F (hch) i o equal t o zero. Center of presoure var ia t ions f o r Mach numbers

o ther than 0.98 w i l l be discussed following the complete range (a = 0 t o

180 degrees) of angle of a t t ack effects . It ahould be noted tha t Equation

28 has been revised s$ight ly from its presentation i n Reference 2 due t o

evaluation of addi t ional t e s t da ta In Reference 13.

Region I11 (a > 90 t o 160 degrees1

Upon close examination of the t e s t data (Reference 13 and 31) i t w s s

found tha t a l inea r var ia t ion could be adopted between a - 90 and 160 de-

grees. Thus, the magnitude of XCp/CR fo r each end condition (a = 90 and 160

degrees) w i l l be required. The value of XCp/CR a t a = 90 degrees, a s pre-

viously s t a t ed , coincides with the panel centroid of area. The value of

XCp/CR a t a = 160 degrees fo r the basic Mach dumber 0.98 can be obtained

d i r e c t l y from Figure 71. Although the value of X /C a t a = 90 degrees is CP R

independent of Mach number, the value a t a = 160 degrees is not. This Mach

number variat ion, while d i f f e r i n g from tha t associated with region I1 (a < 90

degrees), a l so w i l l be forthcoming.

Region I V (a > 160 t o 180 degrees)

s e r i e s approach was used i n l t e u of the graphical type solution of region 1 . 11 (a < 90 degrees). Test data indicate tha t t a i l normal force and hinge ,

moments a r e l inea r from a = 175 t o 180 degrees; thus center of' pressure is

constant. Chordwise center of pressure data fo r a = 175 t o 180 degrees a r e

presented i n Figure 72 and a r e the baeis for the second half (a > 90 degrees)

of the c o r r e l a t i o n u t h o d . A power aerie8 eolu t ion us8 used Lo es t ab l i sh ing

the center of presaure va r i a t ion betweeu the tw angles of a t t a c k (a - 160

t o 175 depeea). Upon exuunination of t he ava i l ab le rest d a m i n t h i e reg ioa

a t h i r d order eerie8 equacion was considered u t i a f a c t o r y and i n the wual

vay boundary condit ions were sought. Magnitudes of XCp/Cp a t a - I60 and

175 degrees were used t o fix both ends of t he curve. The slopes of XCp/Cp

a t these end condit ions, vlz.,

a(x /c CP = 0 a t a = 175'. were used a s the t h i r d end fou r th aa boundary condit ions.

Applying these boundary condi t ions to the fol lowing power aerles:

2 XCp/CR * A. + Al a + A2 a + A j a 3 ' (29)

yielded the equation :

there 0(a ) - 112.75928 - 81.4741 (a) + 14.58789(a2)

C(a) = -32.,57471 + 22.3?%92 (a> - 3.81911 (a2)

The Al(XW/CR) term accounts f o r h c h number e f f e c t 8 a t Pach numberr o t h e r

t h a t h c h 0.98. It ~ h o u l d be noted t h a t Hach number co r r ec t ions are

l imi ted t o t h e term XCp/CR a t 160 degrees i n equat ion 30 because XCp/CR

a t 175 degrees (Figure 72) i r a l ready Mach number dependent. Values of

B(a) and C(a) versus angle of a t t a c k a r e given i n Figure 73. Thus equat ion

30 permits ca l cu l a t i on of X /C a s a func t ion of a , i n t he range from a = CP R

160 t o a = 175 degrees.,

Ef fec t of Mach Number Var ia t ion on XCP/CB

The inf luence of Mach number on the c e n t e r of pressure has been accounted

f o r by two methods. These methods a r e dependent upon the angle o f ' a t t a c k

region, i.e., a < 90 degrees o r a > 90 degrees. This r e s u l t s from the f a c t ' I

t h a t t he cen t e r of pressure is independent of Mach number a t a = 90 degrees.

Effect of h c h Number Var ia t ion a t a < 90'

For angles of a t t a c k below 90 degrees , t h e e f f e c t of Mach number is

presented a s a percent change i n t h e va lue of equat ion

(28). It i r r eca l l ed t h a t t h e bas i c value, which r ep re sen t s t h e

increment i n xCp/CR e x i s t i n g between a = 90 degreas t o any a < 90 degrees,

corresponds t o t he baegc Mach 0.98. The Mach number v a r i a t i o n parameter

F(Mach) of equation (281, which is determined by the measured d i f f e r ence i n

X C p / s between Mach 0.98 and t h e o ther Mach numbers, is shown i n Figure 74 ' ,

a s a funct.ton of aspec t r a t i o . h u e , f o r angles of a t t a c k less than 90

degrees, t h e bas i c XCp/CR value can be modified t o r e f l e c t , the ef f e c t s of

Mach number from Mach 0.6 t o 3.0.

Ef fec t of Mach Number Var ia t ion a t a > 90'

Mach number v a r i a t i o n s of X /C f o r angles of a t t a c k g rea t e r than 90 CP R

degrees a r e accounted f o r i n a s l i g h t l y d i f f e r e n t manner. Ae previously

128

mentioned, the value of 5 /C cp ' b l 6 O in equation (30) 'is not Mach idber

dependent and rnus t be modified ' to include the effect of Mach number. This

I is accomplished by adjusting the basic X /C value at a = 160 degrees as CP R

I f ollows :

I The A1(XCp/CR) term, which is merely an increment applied to the basic

I value of center of pressure at o - 160 degrees, was determined by fairing a e u m e through the measured difference between X& at M - 3.98 'and

M > 0.98 for a - 160 degrees. The magnitude of bl(XCp/CR)br Mach > 1.0 is

I shot& in Figure 75. For Mach * 0.98 no correction is required.

I The effect of Mach number at a - 175 degrees is accounted for: as I shown in Figure 72. Thus, the correction for Mach number at a = 160 degrees

completes Mach numbzr variation for angles of attack greater than 90 degrees.

Use of Method - This section will demonstrate the use of the method in predicting

XCp/CR for angles of attack from 0 to 180 degrees at M = 1.15 where the

physical characteristics of the fin are:

AR = 1.0

a - 3.5 First a general description of the method will be presented. This will be

followed by a numerical example. The results vrll be compared against

experimental data.

1 Calculate XCp/CR a t a - 0 t o 90 degrees: .L

, Using Figure 70 f ind Xcp/Cg a t a - 90 degrere;

b Using Figure 71 find XCp/CR a t desired angle of a t tack; - Calculate (A X /C ) a t the desired angle of a t t ack by

CP Ras9~ 5

using the following expreseion

d Using Figure 74 f ind the function F[kch) a t fhs desired - Mach number;

Using equation (28) ca lcula te XCp/Cg; -

2 Calculate xCp/Cp a t a" 90 t o 180 degrees: - 8 Use value from l ( a ) above fo r XCp/CR a t a - 90 degrees; - b Using Figure 7 1 f ind XCp/CR a t a = 160 degrees; - c Using Figure 72 f ind XCplCR a t a - 175 degrees; .-

d Using Figure 75 f ind A1 (XCp!CR) fo r desired Xach nt~mber; - Calculate xCp/cg a t a = 165 for desired Mach number -

a*f 60 tpWO 98

f Calculate (A c) a8 follo*S -

& Calculate i n i t i a l s lope a t a - 160 degrees;

I n i t i a l Slope - h Uaing Figure 73 f ind B(a) a d C(a) a t desired a; - i Calculate XCp/CII using equation 30 -

3 Using the r e s u l t s of s t eps 1 and 2 combined with XCp/Cg - fo r a = 90 degrees, the chordwise center of pressure fo r a given

f i n can be determined throughout an angle of a t t ack range

of a = 0 to 180 degrees.

Numerical Example

Following are ' the reeul te obtained when the previous procedures a r e

applied :

1 Calculate XCp/CR var ia t ion with a (a = 0 to 90 degrees) f o r - the following f i n geometry a t M - 1.15.

8 Using Figure 70, X /C - = 0.611 CP Ramgo

b Using Figure 71, XCp/CR a t various a ' s up to 90 degrees: -

2 Calculate the remaining variation v l t h a l p h (90 a 5 180 - degrees) at M = 1.15

5 Prom rttp la , - 0.611

b Ualrq Figure 71. XCp/Cg - ( ) -160 - 0.748

Using F i g u t e 72, X /C ( R ) a175 - 0.907

e Calculate XCp/CRat a - 160' for M'- 1.15 -

& Calculate i n i t i a l elope a t a = 160 degrees

In i t ia l Slope - 0.738-0.611 * (70ll80)r J = 0.10395

Using aquation (30) to calculate XCp/CR

Data Comparieons

The rosulte of the numerical example are compared with experimentnl

data in Figure 76. Clcarly,tt w u l d be desirable to show compariaone between '

the reeulta determined by the method and completely independent test data.

Unfortunately, due Lo the lack of such data, comparieone are restricted

to the ex&riwntal data mources that were used in developing the correlation

method. However, the specific test deta used for coiparisons were not

directly used in the conetruction of the method. Additional comparisons

are shown for a triangular fin of 0.5 aspect ratio at aubeonic and traaeonic

h c h numbers. A cornpariron at Mach 0.80 ip ehonr in Figure 77 and Figure 78

shows a comparison at Mach 1.30. Agreement ie quite good throughout the angle

of attack range For all comparisons. Some deta scatter te noticed near the

extreme ends (a 0 and 180 degrees) of the angle of attack range. As m y

be expected, s c a t t e r of t h i s type usually reeu l t e from the order of

l~sgni tude e f f e c t s associated with very emall forcee and moments ured in

determining the center of pressure location, Zn general, the corre la t ion

leth hod agrees within 2 .5 percent of t he experimental data with the possible

exception of a few i so la ted areas, These a reas usually involve only a very

small segment of t he ca range such a s shown i n Figure 77 near a = 40 degrees.

A deviation of approxlnateLy 3 percent vae noticed from a Q, 130 to 150

degrees f o r the f i n i n Figure 76.

Figure 69. Chordwiee Center of Preeeure Variation to 180 Degrees

0 0.2 0.4 0.6 0.8 1.0

TAPER RATIO (h)

Figure 70. Chordwise Center of Pressure Variation with Taper Ratio at Alpha o f ' 9 0 Degrees

Xc? 4;-

Figure 71. L a i c Curve8 for Xc,/cR a t Reference klch Number 0.98

(0 to ldO Degrees)

Figure 72. Baeic Curves for XCp/cR at Reference Angle of Attack 175 to 180 Degree8 (H - 0.6 to 3.0)

160 165 170 175 ANmE OF ATTACK (a) *DEGREES

ANGLE OF ATUCI; (a) *DEGREES

Figure 73. Pwer Sariem Co~otante vereue Angle of Attack

Figure 74. Uach Number Correction Factor for a < 90 Dqraer

, CORRECTION WE TO M ~ 1 . 0

mmi HmiBER

Figura 75. Variation of Al(XCp/CR) With Mach Number a t Upha of 160 Degrees

1.0 0

TAPER RATIO - 0

0.8

0.6

0.4 - P8EDICTIal m D

0 (I- nn)

0.2

0 1 + -20 0 20 40 60 80 100 120 140 160 180

Amaa OI A T r m (a)* D B ~ I S Figure 77. Comparison of Predicted and Experimental Center of

Pressure Loeati6n (X /Cp). Mach = 0.80 C P ~

Figure 78. Comparieon of Predicted end Experimen' il Ceater of Prerrure Location (X /CR), Mach = 1.3::

C P ~

5.2 Body-Tail Confisurations

5.2.1 Tail&-Body Normal Force

S-ry

A method is presented t o p red ic t CN , the normrrl force Coefficient on T (B)

the hor izonta l , undeflected t a i l s of body-tail configurations. The method

i a applicable to "plus" configurationcr a t Mach numbers between 0.6 and 3.0

and angles of a t t ack from 0 t o 180 degrees. The method consie ts of a pro-

cedure f o r ca lcula t ing an in ter ference fac to r , RT(B), which can be applied

to i sola ted f i n data o r the r e s u l t s of Section 5.1.4 t o determine t a i l -

on-body normal force coef f i c i en te , CN* (B). Aerement between predicted and

experimental r e s u l t s were found t o be qu i t e good.

Background

The normal force on a t a i l f ixed t o a body d i f f e r s from tha t on an

isola ted t a i l a t the same angle of a t tack. This d i f ference is a t t r ibu tab le

t o the in ter ference of body-induced upwash and lee-side vortex downwash 0'1

the t a i l flow f i e ld . To predict tail-on-body normal force, i t is necessary

t o correct i sola ted f i n data fo r theee in ter ference e f fec t s . Methods a r e

avai lable which predict each in ter ference term separately (Reference 17) o r

combine the two i n t o a s ingle in ter ference fac to r (Reference 3). However, these

methods a r e not applicable over the e n t i r e angle of a t t ack (0' - 180') and Mach ,

number (0.6 t o 3.0) ranges. The method of Reference 17 is l imited t o angles of

a t t ack below tha t a t which the body lee-side vortex pat tern becomes asymmetric

(a < 30'). In i t s present form, the method of ~ e f e r e n c e 3 is limited to

angles of a t t ack l e s s than 60 degrees and to transonic Mach numbers.

Method Development

Due to the complicated nature of the flow f i e l d an analyt ic approach t o

methad development was not considered. An empirical approach was selected.

The data of Refersnce 13 were insuf f i c i en t t o d is t inguish the contribution

of each type of interference t o the t o t a l ; therefore, an extension of the

method of Reference 17 was not pract ica l . The nature of the instrumentation

used t o co l l ec t the data of Reference 13 did provide euf f i c i en t information

t o ca lcula te the t o t a l in ter ference a s the r a t i o RT(B) = (tail-on-body

normal force C ~ ~ ( ~ ) / t a i l alone normal force C N ~ ) . These data could be

corre'lated and presented i n a form l i k e tha t of Reference 3. However, the

resul tant method k u l d be awkward and time consuming t o use. In order t o

develop a rimple, easy t o uee preliminary design tool , a power s e r i e s approach

t o method development was selected. In the usual way boundary conditions

were sought. A s indicated i n Reference 3 the value of %(B). a t a = 0 degrees

can be s e t equal t o the value' of predicted by potent ia l flow the#:-y.

Values of $(B) a r e presented i n Reference 30 but f o r the sake of completc-

ness a re preeented again here i n Ftgure 7.9, As a second bouniary condition,

' the value of % a t a - 180 degrees can be assumed equal to 1.0 i n the (B)

absence of any forebody effects . A aurvey of % data (Reference 13) 0 )

versus angle o f m a t t a c k yielded fu r the r boundary conditions. A t a = 30

degrees, the value of was observed consis tent ly t o be 1.0 with

a% B equal t o Zero* It was a180 noted that a t a = 130 degrees. the value aa

of %(B) i n Figure .79 with %(B) again equal t o zero. The value of aa %I)

a t a - 90 degrees was taken a s a f i n a l boundary condition. The data showed

tha t the value of a t 90 degrees waa not constant; therefore, i t was

l e f t a s a f r e e variable, % ( s ) ~ ~ , *

Applying these boundary con l i t ions t o the following power s e r i e s ex-

pans ion 3 5 - a + a a + a2a2 + a a + a4a4 + aSa + a g 6 %(B) 0 1 3

yielded

3 4 %(,, - (3.9808Oa - 3. 67990a2 - 1.95129a + 3.376380 - 5 6

1.32994a + 0.16987a ) + (1-7.34322a + 20.55753a2 - 27.317r7a3 + 17.644470~ - 5.28848a5 + 0.58856a6) 5 +

(8) 4

(3.36248a - 16.87764a2 + 29.27176a3 - ' 21.02285a +

vhich can be r ewr i t t en a s : ,

Values of Ao, A1 and A a r e p lo t t ed a s a funct ion of angle of t*'

2

a t t a c k i n Figures 80, 81 and 82.

Corre la t ion of the ca lcu la ted va lues of RT(B) showed t h a t t h i s 7I/2

quant i ty varfed with both Mach number and t a i l t aper r a t i o . Values of

RT(B),,~ arc presented i n Figure 83 a s a func t ion of Mach number and taper

r a t i o a s obtained by f a i r i n g curves through the t e s t da t a of Reference 13.

In t he course of checking r e s u l t s predicted by Equation 31 aga ins t

experimer,tal da t a , a problem was encountered both subsonice l ly and t rans-

onica1l.y f o r angles of a t t a c k between 0 and 30 degrees. The .va r i a t i on i n

%(B) w:th angle of atta:k a e predfcted by Equation 31 was much more rapid

than tht! experimental d i ~ t a tended t o ind ica te . To account f a r t h i s , a

second power s e r i e s was used t o develop a method app l i cab l e t o subsonic

and t ransonic Mach numbers over t h i s range of angles of a t t ack . A s before,

t h e va lues of % ( B ) at a - 0 and 30 degrees were taken t o be 5 and 1.0, ( B )

respect ively. A s a t h i r d boundary condi t ion , i t can be shown t h a t a % ( ~ ) = aa

zero a t Q - 0 degrees.

Applying these boundary condi t ions t o t he following power series ,

expansion

yielded

' which can be r ewr i t t en ' a s

where

A. - 3.64756a2

A1 = 1 - 3.64756~~ 2

Values of A. and A1 a r e a l s o included i n Figures 80 and 81.

Use af Method

A genera l de sc r ip t i on of how t o apply t h i s method w i l l be presented

i n t h i s sec t ion . This w i l l be followed by a numerical exampbe i n which

R~ (3) w i l l be ca lcu la ted and applied t o i so l a t ed f i n da t a t o determine CN

T (fi) '

1 U s i ~ g Figure 79 determine the va lue of % at t h e - ( B )

appropr ia te value 02 d/s.

2 Using Figure 83 determine the value of RT(g) - a t t he appropr ia te ~ 1 2

Mach number and t ape r r a t i o .

3 Uring the r e r u l t r of r t ep r and 2 and Figures 80, 81 and 82 - apply Equation 31 (for rubronic and transonic Much numbers

u re Equation 32 f o r angles up t o 30 dogreee) t o ca lcula te

valuer of %(B) f o r angles of a t t ack between 0 and 180 degrees.

4 To determine the normal force coef f i c i en t s fo r a t a i l fixed - t o a body, multiply i sola ted f i n data or the r e s u l t s of

Section 5.1.4 (p. 91 f f ) by the valuer of RT(B)*

Numerical Example

Calculate f o r the following body t a i l configuration a t M = 0.6.

Body:

& &N .;i = 10.0 - = 3.0 d d - 1.25 in.

T a i l :

Using Figure 79 o r d/s - 0.3

Using Figure 83 f o r M = 0.6 and X - 0.5 .

For M = 0.6 apply Equation 32 fo r 0' < a 2 30' and Equation

31 f o r 30" < a 5 180'. Use Figures '60, 81 and 82 t o determine

general coe f f i c i en t s A A and A?. 0' 1

1.222 j Equation 32

1.139 \

b Equation 31 I

4 Using isolated ftn data obtained from Reference' 13, calculate C - N~ (8)

4 (Continued) -

Data Comparisons

The results of the numerical example are compared against experimental

data in Figure 84. Further comparieone for a variety of Mach number* and

tail geometries are presented in Figures 85-89. Agreement is quite good

in all cases.

BODY DIAMETER d SPAN B

Figure 79. Ratio At Zero Angle of Attack

ANGLE OF ATTACK-DEG.

Figure 80. General Coefficients For Calculation Of RT 0)


Figure 82. General Coefficients For Calculation Of R.r (B)

Figure 83. Interference Factor A t Angle of Attack at 90 Degrees

bod9 Tail - 0 Exporbentd (hf. 19) t/d - 10.0 i - 0.0 - edictad ad

Figure 89. Comparison O f Experhental And Prcdlcted Results, C , - 0.8 *T CB)

5.2.2 Tail-to-Body Carryover Normal Force

Summary

A method is described t o predic t IBb), the tail-to-body carryover '

normal force coeff ic ient . The method app l i e s over the Mach range 0.6 - 3.0

a t angles of attack from 0 - 180 degrees.

Background - When load-carrying l i f t i n g surfaces a r e fixed t o a body, loadi t f a l s o

appears on the body due t o carryover e f fec t s . The normal force thus

generated is denoted here by I B ('r) (see Sectlon 4.0). In potent ia l flow near

zero angle of a t tack, I B(~) reduces t o which is determinable by

l inear ized theory (Reference 30). Use is made of KBtT)in the present method.

Method Development

Separate methods a r e presented f o r the transonic (0.6 2 M 2 1.3) and

supersanic (2.0 ( M 5 3.0) regimes, respectively. Interpolation should be

used fo r Mach numbers between 1.3 and 2.0.

Transonic Mach No.:

The general form of the I B (TI

curves, a s derived from experimental data,

is shown schematically 'n Figure 90. Three major values a re used i n power

se r i e s deve.l.opent, Ia, Ib, and 1,. Note tha t a t zero and 180 degrees angle

of a t tack $,(,..is zero. The basic- p o q r s e r i e s fo r portforis A and B of the

The dlvfslon of the angle of at tnck range and the points med as bounciary

corditfczs a r e chosen by oSsrrrat lon of the trend in the t e s t data.

Boundary conditions are:

Substitution of these conditions into the power series yields:

Portion A (a in radians)

IBCT) - C0.172 Ib + 2.562 I,] a

' I - c0.394 Ib + 1.930 I,] a 2

+ [o. 353 1, + 0.226 Ib] a 3 '

Portion B (Using only first 3 terms in series) (a in radians)

I B(T)

- 19.592 Ib - 39.869 1, + C30.271 I, - 13.286 I ~ ] a (34)

+ [2.2441b-5.596 IC] a 2

Correlation of data can then proceed with attention concentrated on

I,, Ib, and Ice

Ln general, Ia, Ibp and 1, are functions of coafiguration geometry and

flow eonditione, i.e.,

= I Iasbr~ a,b,c ( A p A% d/s, M)

The aesumption is made that the variables are separable, so thkt the

and the form of each reparate function i r obtained by examination of the

exper i l~anta l data. Sooletimer corre la t ion doer not require use, o r complete

wparati'on, of a11 the variable.. For oxample, the following representat ions

of I,, Ib, and IC were found t o be ru f f i c i en t :

F (W F.(H) I, - I. k r i c . 5 - Ib (ARB n )

F (AR) = c - 1 c B 8 s i c c

Ewrinat ion of the t e a t data of Reference 13 rhomd tha t corre la t ions

of the quan t i t i e r I., 5, and Ic made i t por r ib l r t o generate boundary

condition8 f o r the I B (T)

function which leadr t o good agreement between the

model and the t e a t data.

The corre la t ions c f I., I,,, and I a r e prerented i n Figure 91a, b, C

and c.

Ure of Uethod (Transonic1

S?lppore it i r required t o ert imate the tail-to-body carryover normal

force fo r a configuration a s f o l l o w :

Tail : AR = 0.5 A - 0.0 d / r - 0.5

M - 0.8

From Figure 91:

? (AR) r a m ) a a c a

* 1.0 x 2.6 x (1.65 - 0.5) - 2.99

Ib - 0

? (AR) I C - =c Basic c - 0.3 x 2.5 - 0.75

Hence, t h e Equations (33) and (34) become:

' ( 3 U ) (a i n rad ianr )

(34A)

The r e s u l t s a r e compared with experimental da t a i n Figure 92 . It w i l l

be reen that matching i r q u i t e good.

S u ~ e r r o n i c Mach No.

For per sonic Mach numbers t he procedures f o r ca l cu l a t i ng I B(T) are

genera l ly rimpler than f o r t he t ransonic care , However the Ig(T) curve is

divided i n t o three , r a t h e r than two, p a r t r and is rhown rchcmatical ly i n

Figure 93. This i r the form t h a t t he t e s t da t a takes and the curve

r e p r r r e n t s a f a i r i n g of t he da ia .

The t h r e e ~ j o r por t ions , A, B, and C a r e rhown, along wi th t h e

important c o r r e l a t i o n inputs , 11,,12, and 13. The following representa t ions

a r e ured.

A: I - Il s i n a B (TI

- 1.46 I2 - 3.076 I., ] a2

(35) a , i n

(36) I degrees

(37) ' a in ' degreee

The l a e t equat ion was obtained from the usual s e r i e s representa t ion ,

with boundary condit ions:

Again, using repara t ion of va r i ab l e r ,

These q u a n t i t i e s a r e presented i n Figure 94a, b, and C, respac t ive ly .

Uae of Method (Supe raon ic~ --- Suppose i t i r required t o o r t ima te t he tail-to-body carryover normal

fo rce f o r 8 conf igura t ion a a follows:

T a i l : AR - 2.0, A - 0.5, d/r - 0.3

M - 2.5

From Figure. 94.. b, and c : ,

* l * I1 Bad.'! ? ' 1 (dl.) - 0 . 7 5 x l . O - 0.75

I2 - I2 Basic F 2 (M) FZ(d/r) - -a x 1.0 x 1.0 - -2.0

Portion A, IB(T) - 0.75 ain a

Portion 1, IMT) -0.75- (0.75+2.0) ,/F

Portion C, - -33.515 + 24.6720 -4.4574a 2 IB(T)

A comparison between prediction and experiment i r shown in Figure 95.

It will be men that urtchiag i r quite good.

Figure 90. Tranaonfc I B ( ~ i , Schematic

NWY DIA/OVERALL SPAN TAIL ASPEm RATIO

Fimre 91.. Curve. for Entimation of Transonic I (Al l 1) .

T i y r e 91b. Cl~rves for Estimation o f Transonic I,, (All A and d/a) '

l i . ~ u r e 91c,. Curves f cr C a t i u t Ion of Transonic I (All k and Mach Ntmbrrs) C

0 Experiment mef. 13) - Prediction

2

ANGLE OF ATTACK-DEC.

Figure 92. Comparison Between Predicted And Experimental IBT

5 ,.--- ----------- ANGLE OF ATTACK '?M;REES

Figure 9 3 . Supersonic T v Schemat f c

0 $ 5 1:0 0.3 0.4 0.5 TAPER RATIO WDY D I A / O V E W L Sow

Figure 94.. Curver for Emtimation of Superronic XI

- 0 0.5 1.0 TAPER PATIO

7

2 .O 2.5 3.0 MACH NUHBER

7 - 0.3 0.4 0.5

BODY: 01 AIoVERALL SPAN

Figure 94b. Curver for Ertimation of Supcrsoaic I 2

0 0:s 1.0 TAPER RATIO

0 - 1.0 2.0 ASPECT RATIO

Figure 94~. C U N ~ S for Estimation of Superronic I3

5-2 .3 Tail-To-Body Carryover Normal Force Center of Pressure

Sumrary

X method 1s described t o predict XCp , the center of pressure of I (T)

norrral force carr ied over t o a body from horizontal t a i l s . The

method is an extension of an ex i s t ing technique. i t i s val id f o r angles

of a t t ack from 0 - 180 degrees i n the Mach range 0.6 - 2.0

It should be noted that XCp is t h e a x i a l distance from the missi le 1 (T)

w e e t o the point of application of the carryover force. The same point ,

located r e l a t i v c t o the rruface leading edge i e defined a s Xcp B(T) *

Background

Aa ex t r t ing method, Reference 4, which appl ies to the angle range

0 - 90 degrees a t Mach numbrrr from 0.8 - 1.2 was available a s a s t a r t i n g

point. The~formulationr of chis method were such that the procedure was

e a s i l y extendable to the angle range 0 - 180 degrees and the Mach number ,

The o r ig ina l mathod was based pattXy on the theoretical r e s u l t s of

Reference 30 from vhich value; of %p' B (TI

near zero angle were obtained,

These r e s u l t s were used o r l g i m l l p fo r tha condition of a t e f l with no

afterbody. k -ve r , when sale of crttqck has reached 180 degrees, the

l i f t i ~ g suri.c@ does have an "afterbcdp" and the Reference 4 method was

laoctified t o r e f l e c t tt.ie.

Since the boundary eonditionr on the curve of ( q p B R l / C R ) vetsua

angle of a t tack a re eaai ly datetmined, a power seriecl approach t o cot re la t ion

w a l *laed.

Boundary conditionm are:

a - O* ('LEpB /CRIo given- by Reference 30 (see below)

0 ( l i n e a r theory)

= 0.5 (mid-chord po in t )

= 0'

given by Reference 30 (.see below)

a - aa & P ~ ( ~ ) / C ~ ) , , ' 0 ( l i n e a r theory)

It is shewn i n Reference 30 that curves of (XCPB(T)/CR)o, the load

loca t ion near zero angle of a t t ack , can be constructed f o r various rad ius /

semi-span r a t i o s on the b a s i s of BAR f o r conf igura t ions with a f te rbodies .

It is noted tha t the r a t i o a l p used i n Hefarence 30 is equivalent t o (d/2)/

b / 2 ) o r d / s i n the terminology of t h i s repor t . These curves a r e presented

i n Figures 96a, b, and c. The major reason f o r t h i s representa t ion i s t o

uae the s lender body theory r e s u l t s f o r BAR - 0. Reference 30 ind i ca t e s

t h a t the same representa t ion can be employed f o r conf igvra t ions with no

a f t e rbod ie s a t supersonic speeds, but does not present the a c t u a l curves.

It does, however, p resent information (Chart 14b of Refererce 30) which can

, be used t o cons t ruc t p a r t i a l l y the curves of (XCpB(T)/CR)o away'from t h e

BAR - 0 poin t . Chart 14b presents d a t a on load loca t ion a8 a funct ion of

Bd/CR which may be wit ten' i n terms of 6, aspect r a t l o and body radius/semi-

span r a t i o i f t a i l t aper r a t i o is known. For example, f o r t r i angu la r planfonn

t a i l s with no t r a i l i n g edge sweep, the equa l i t y d/CR - AR/?(p/a-1) holds.

From r e l a t i o n s l i k e t h l a , Chart 14b was converted t o Figure 97a, b, and c of

t h i * r epo r t i n which valueai of &B(T)/CR)o a r e presented. In Figures 96

and 97 the va lues of load loca t ion a t BAR - 0 were taken from slender body

theory under the assumption t h a t no t a i l f o r ce was developed a f t of the

maximum span. Values f o r a /p - 0 ( t a i l with no body) were taken from

supersonic wing theory.

For subsonic speeds, Figurzs 98 and 99, taken d i r e c t l y from Reference

30, may be k d .

Rather than at tempting t o f i t a s i n g l e equat ion t a t h e e n t i r e angle of

g t t ack range, i t was divided i n t o two eec t ions , 0-90 degrees and 90-180 degrees.

'ihe fun~:tlonal. Corm chosen m p i r i c a l l y

X,"

represent

Uhen the f i r s t four of t he boundary condit ' ions defined above a r e used, the

se;ies takes the fo l loyfng fo r s .

(a i n rads)

The v a h e of k p B(T)/CR represents , i n non-dimensionnl fot~, t he

d i s t ance fro.? the f i n root chord leading edge t o the cen t e r of pressure of

the load generated on the body due t o the presence of a t a i l . The second coef-

f i c i e n t 2a3/ (n/213 - 3aZ/(n/2)' has been evaluated between 0 and 90 degrees

and is shown i n Figure 100.

90' - 180. By now measuring the force location from the trailing edqe ( Q p

B(T)) and definfng the quantity iT = n-a , it is possible to derive an expression directly analogous to Equation (38) by utilizing the last four boundary

codit iona .

where is meamred from the tail traizing edge and

Use of Mkthod

Uee of the method for predicting Xcp will be demonstrated in B (TI

conjunction with the other methods required to predict the center of pressure

of a complete body-tail configuration. Initially, the method for predicting

, XcpB(~) will be described generally, Then a numerical example will be given

of XcPs(r) cal~ulatio~, plus the-other calculations necessary to predict

centera of pressure on a complete configuration.

1 Calculate XCPB(T) for 0' 5 a 5 90' using Equation 38. - a Depeading upon Mach number, determine -

xCPBCT)I uSia either Figures 97 or 98.

CR 0

b Using Figure 100, determine values of - +Y *

2a3 - 3aL - - at selected angles of attack.

- 2 Calculate XCpB(T) - for 90' j a 2,180. uring Equation 39A

a Depending upon Mach number, determine XcpBtt) 1 - uring

CR I n either Figures 96 or 99.'

3 2 b Ueinq Figure 109, determine values of & - - -

where o - 180 - a

Numerical Example

Calculate the centers of prarrure at N - 0.9 for r body-tail I '

configuration having the following ihcractaristicr:

Body: Tail :

d = 1.25 inches

1 C.lcul.ta CN - B a ~t tranuonic k c h numberr, uae tho method o t ~efardnce 12 -

for 0' 5 a 5 40'

u8e the method of Section 5.1.1 (p. 39 f f )

2 Calculate XCp udng the method of Sectlon 5.1.2 (p. 61 i f ) - B

3 Urlrw the method8 of Soctlon 5.1-4 (p. 91 ff) a d 5.2.1 (pi 143 ff) -

4 thing the method of Section -

5 U m i q tbe r t & d o f Section 5.2.2 @. 161 i f ) , Calculate I - - z-/ - 0 (T)

x c ~ 6 U8iq tho procedure outlinod above, calcul8to I (Tl - d

X c ~ where I @ i I d

2 for a 5 90. d

7 TO calcu1ata Q P B ~ , . the center of prammure of tb bodytai l d

combination, apply tho ramultr of ' ~ c a ~ r 1 6 to the following equation:

X C P ~ ( ~ ) S~

'T(B) - + I ~ ( ~ ) d

d b B T - - ref

d .

Data Coaoarimonr

Thr 'rceults of this t e s t case (C and X ~ ~ B T ) are compared against experimental N~~ - d

data i n Figures 101 and 102. Good agreement is obtained betwcen the predictions

end experimental data.

--

1 2 3 4 5 - 6 7 KFFECTIVt ASPECT RATIO.IUR

Fi~virr 96. Curves for Determining kpB(T) /CR With Af tarbodiea

for Superronic Specdm

I I

0.6 I

9

0.4 . ., -

0.2 '--+ I (a) XaO

o* i A- tl

Reference 30

EFFECTIVE ASPEm RATIO , Figure 97 . Curves For Determining XCP~(~)/CR for No Afterbodlee at

Supersonic Speeds

Reference 30

3 4 5 6 7 8 EFFECTIVE ASPECT RATIO, t3AR

Reference 30

, EFFECTIVE ASPECX RATIO, BAR

Figure 99. Curre* for Determining Xcpg (T) /CR for Subsonic Speeds (Zero Trailing Edge Sweep)

Firrure 100. Coefficients Required for Evaluation of

Body Tail

1 -0.5 . 0 Experhntal h f . 13) LN/d = 2.5

AR 1.0 - Predicted a,/a = 7.5 d/s - 0.5 d - 1.25 Inchem - Cg - 1.667 Inchem

Figure 101. Conparison Between Redlcted And Experimental Data, C

N~~

w e - - - - AR - 1 . 0 r*/d = 7.5 0 bp;crimental ( h f . 13)

d/. - 0.5 d - 1 .25 Inchen - R e d l c t e d CR - 1.667 l a c h e s

5.3 Body-Strake-Tail Configurations

5.3.1 Inc rewnta l Normal Force Due t o Straker

Stmmry

A mthod is presented for ertimating the t o t a l incrmental normal force

coeff ic ient , A%s

,due t o low aspect-ratio serakes on a slender tangent-

# \

ogivocyl iader body a t a r o l l angle of zero (+ orientation). Thie method

\ covers angles of a t t ack up t o 180 degrees and a Mach number range of 0.50

to 2.2 .ad r.pra8antr 83 extenmion of an exidLing l w a n g l e technique.

Background

The idd i t ion of s t raker t o a body produce@ an increased normal force

which i r a function of s t rake r i z e r e l a t i v e ' t o the body and s t rake aspect

ra t io . The incremental normal force may be estimated a t low angles of

a t t ack from Section 4.3.1.2 of Reference 17. A t higher angles no methods

ex i s t for calculating the increase. This section describes the construction

of such a method. The data forming the b a d e for correlation were obtained

from t e e t s on a par t icular USAF miselle design. Since the strakes used

were not instrumented t o record normal force, t he follcning formulation was

6CNBs waa datemined d i rec t ly from t e e t data a t Mach 0.6. 0.55. and

1.2. Due t o a lack of body plus s t rake data a t zero r o l l angle, valuer of

A Q B s a t Mach 1.8 and 2.2 were derived ueing available t o t a l conf lguration

snd body alone data a t those Mach numbers i n cocjunction with a factor from ,

Hach 1.2 data defining t e l a t l v e t a i l and atrake contributions. A curve-

f i t procedure was used for data correlation.

Uethod Development

W i n a t i o n of data avai lable a t f i v e Mach numberr (0.60, 0.95. 1.20,

1.85, and 2.2) revealed nebera1 fea tu res uneful i n curve f i t t i n g (see

Pigura 103). A curve of ACN versur alpha a t each nieh number exh ib i t s BS

peak8 of approxlnutely equal magnitude a t a - 57. and a - 135.. The value

of AC a t these peaks is Mach number dependent. Between a - 80- and N~~

a - 120.. the value of A% o s c i l l a t e r about a mean value which i r independ- BS

at of Unch number. The rlopeu (ACN ) a t a - O* and a - 180. appear t o B S m

be about equal i n magnitude but with opposite signs. A pomr s e r i e r formu-

l a t i o n urine the afotemantioned curve q u a l i t i e s a s boundary conditions 'was

the approach selected t o f i t a general curve t o the data.

A third-order power ae r i e s of the form

2 - a + a a + a2a + s3a 3 0 1

was umad with the following boundary conditions:

where

Ac~Bs - 0 a t a - 0' and 180'

ACN - J1 a t a - O*

BSa

ACN - -J1 a t a - 180'

"a

AC - K a t a - 57.3' and 135. N~~ a

ACN - t a t a = 80. and 120.

BS

S

( K ~ ( ~ ) , + I(y ) (-L) (: AR ) [Reference 17 . (*) 'ref s

Section 4.3.1.21

, K - A peak value a t o - 57.3. and 135. N~~

The bracketed term i o an ~ p i r i c a l corre la t ion preeented i n Figure 104

L - man va:ue of AC , '80. 5 a 5 120. *ss a

S, - s t rake eingle span exposed area

'ref - Reference area f o r 4QBS ( 8 ~ ~ 1 t o body cross r sc t iona l area)

SSB - Area of two r t r aker + planform a r e a . o f body between' , ,

et r rkee

Note tha t the boundary condition J1 ha8 been generalized by the presence of

aspect r a t i o , Kg(W) and Q(B) , the l a t t e r t o be a function of d! a. Planform

area was found t o be an e f fec t ive corre la t ing parameter fo r the quan t i t i e s

X,and I,.

To aimplify the power eer iee eolution and improve the ercutacy of the

e e t h a t e , the power s e r i e s wae formulated for three 1,ntervale: 0 5 a 1 8 0 ' .

80' - < a - 4 120°, and 120' 5 a 5 180°. Solution of the th i rd order power

s e r i e s yielded ao, 81, a2, and a ae function8 of J 3 1' Ka, and L for the

three angle of a t t ack ranges. Upon eaparating terms, an equation of the

form

wae derived. Equatione for A AZ, and A3 are ae followe:

(0- radians) ,

( a d radians)

Values of A1, A2 and A have been plotted versus angle of attack (Figure 105) 3

to facilitate use of this method. Peak values have been Ss+s/saea

determined empirically and are plotteu versus Mach number in Figure 104.

Use of Method

The method ia used a8 follows:

Given a tangent-ogive-cylinder body with low aspect ratio strakes of the

following characteristice:

body diameter - d

+ d 2 body reference area - - - 4 ref

strake oingle span exposed area =

strake root chord - C ~ s

s t rake arpect r a t i o - atrake exposed eemi-span - b/2

Proceed thus : S

1 compute .I1 - (SOO + \(B)) (+)(I %) l i e f . 17, Seci. 4.3.1.21 - ref

Find ( 1 From Figure 104 fo r the desired Mach number. ss+~/Sr, t

Ka - * 'S+B Compute Ka = -- where SS+B - (2*Sa)+(C * d) S+B ref R~

4 Look ap A1, A*, and A3 f o r the derired angle of a t t ack i n - Figure 105.

5 Subrt i tu te i n the relat ionship -

Numerical Example

Calculate A Q B S a t Mach 0.85 for a body - r t r ake combination having

the following properties:

d = 3.667 in. AR8 - 0.040

'ref - 10.56 sq.in. b/2 - 0.40 in.

S8 - 8,. 06 rq. in . C%

- 14.33 in.

1 from Ref. 17 KB(y) - 1.43, - $(B) -

2 From Figure 104, -0.66- 1tMach0.85 S~+~/Sre l l :

'S+B - (2*8.06) + (14.3Y3.667) - 68.67 8 q . h .

68*67) = 3.00 3 L = 0.461 (- - 10.56

From Figure 104

Data Compflrisons

I n Figure 106 t h e r e s u l t 8 of t h i s method a r e p l o t t e d along with those

d a t a used i n formulat ing t h e method. It can be reen t h a t the power s e r i e s

s o l u t i o n y i e l d s good c o r r e l a r l o n with t e s t d a t a a c r o s s the Mach range

t e s ted . A l a c k of independent d a t a i n t h e des i red high ang le of a t t a c k

range p r e v e n t s , f u r t h e r comparisons.

60 80 100 120


Figure 103. AC General C u r v e Form *BS



Piguie 105- Coef f?cientr for C a l c u l a t i n g hCN BS

Figure 105 (Cont .) . coofficienta for Calculating A$ ,

BS

5.3.2 Center of Pressure f o r Incremental Normal Force Due t o Strakes

S-rs

A method is presented t o predic t XCp , the e f fec t ive center of AJ3S

a pressure of the incremental normal force (ACN ) due t o & low aspect-rat io BS

etrakc on a slender tangent ogive-cylinder body a t a r o l l angle of zero

(+ orientat ion). Thje method covers angles of a t t ack up to 180 degrees and

a Mach number range of 0.60 t o 2.20 and repreeents an extension of an

exis t ing l o r a n g l e technique.,

Backaround

The addit ion of s t rakes t o asbody produces a change i n the center of

pressure locat ion wldch i e re la ted t o the s t r ake e f fec t ive center of preeeure

location, XCp , the s t r ake normal force coeff ic ient including carryover, 4BS

*%BS , and the body alone normal force coefficicut , 5 , and center of

B pressure, XCp . The r t r ake center of pressure locat ion may be estimated f o r

B low angles of a t t ack by the methods of Section 4.1.4.2 of Refe rqce 17. The

prerent work describes the forrrmlation of a method f o r predict ing s t r ake

center of pressure. location at angles of a t t ack up t o 180 degrees. The data

fotming the basta f o r corre la t ion were obtained from tests on a pa r t i cu la r

USAF n i s s i l e design.

Since the s t rakes tested were not instrumented f o r cepter of pressure

determination, the following equation was used f o r thc smnmatfon of moments:

XcprlBS represents the center of pressure of the e n t i r e s t rake normal force

contribution, including interference e f fec t s , and was determined d i r e c t l y

from t e s t data a t Mach 0.60, 0.85, and 1.2, Due t o a lack of body plus s t r ake

data f o r zero r o l l angle a t Mach 1.8 and 2.2, values of X were derived "ABS

using avai lable t o t a l configuration and body alone data a t those Mach numbers

I i n conjunction with a f ac to r from Mach 1.2 data defining r e l a t i v e t a i l and

I

I s t rake contributione. A curve-f i t procedure was used fo r data correlat ion.

Hethod Development

Figure 107 shows the general form of a curve of XCp versus angle of ABS

I a t t ack a s derived from t e s t data. Thie general curve shows tha t Xcp ABS

moves from its a = 0' locat ion t o a point near the etrake centroid a t , C

a s 30". then moves forward a s a+60°. A t a % 120°, XcpABS a t t a i n 8 'its

fa r thes t a f t posit ion, from which it moves forward t o a point near the i' . . centroid a t a s 180'. Center of pressure location8 a t 0, 60, and 120 degrees

exhibited a dependence on Mach number., A power s e r i e s formulatron using

these curve q u a l i t i e s a s boundary conditions was the approach selected t o

f i t a general curve to the data.

XcpABS wae considered t o be a function of Mach number and s t r ake

I geometry. Since the s t rake teeted had two d i s t i n c t r e g e n t s , the area

I centroids of the forward portion (s) and of the a f t portion (zB) were

incorporated along with the a rea centroid location of the e n t i r e s t rake (2 ) S

and the etrake root chord length (CE ). Figure 108 i l l u s t r a t e s the ~ t r a k e S

parameters used i n t h i s analysis .

The equation f o r the apparent location of the incremental force due

t o the addit ion of a etrake ia:

where XCp /d i8 8 function of anple of a t t ack and Mach number, and XLE is S

the a x i a l distance from the body nore t o the leading edge of the etrake.

W t e tha t XCp /d represents the center of presaure of the atrake t o t a l nonnai S

force (ACN ) and is measured from the leading edge of the atrake root , BS

whereaa X~pdas i m umured from the body nom. A mecond-order p n n r merit.

of the form

XCP, - = a. + a a + a2a 2 d 1

was used with the following boundary conditions:

where XCp = center of pressure at a = 0" [Ref. 1 7 , Section, 4.1.2.21 L.

3~ - d 0. 25CR

I- for M < 1.0 d

s - - for M 2 1 .0 d

A .review of the test data suggested che following formuletiora.

/ I, -.. I .. .• - .: . /

/( 'A.

R CP location at a - 60°

xS xS xA" -d - 2 ( '- d -d)

T - CP location at a-1200

KB (cR Sx+ -b7dd

where J2 anr K b are functions of Hach number.

Note that the equations for R and T have been generalized by the presence of

the terms XA, X1, and X, and that, for the limiting case of a rectangular

strake,

XA 09 XB XSP and XS 0.5 CRS

To simplify the power series solution and improve the accuracy of the

estimates, the series was formulated for three intervals: 0 < a, _ 600;

60° < a < 120"; and 120* < a < 180C. Solution for the second order power

ýseries coefficients yielded a., a1 , and a2 as functions of xCPSol -, R, T,

d d

and S. Upon separating terms, a function of the formd

x xcPS CPso S xS- A A + A (-) A (R) + A, (T) 4 A (--)1 I d A 2 d 3 5 d

was derived. Equations for A1 , A2, A3 , A4 and A5 are as follows:

205

(a - radians)

(a - radian.)

Values of An have beer. plotted as a function of an& of attack in Figure 109

to faci l i tate use of this method. Peak value factors J2 and I$ have been deter-

mined empirically and are plotted versus Mach number in Figure 110.

Use of Method

The mthod ir ured as followr:

Given 8 tangent-ogive cylinder with lat aepect-ratio strakes of the folloving

characterlrticr:

body dimtter - d rtrake toot chord - C R ~

etrrke leading edge station - kE - distance from LE to eelpaant A centroid -

diatame from LE to segment E centroid - ZB di*taca tram LL to net .trW centrofd - %

Procaed t hur :

.& determine XcpSO (-0.2SCES for )I < 1.0; - Ts for n r 1.0)

[Section 4.1.4.2 of Ref. L71, I I

2 determine J2 and 16 for the appropriate Uach nuaber <Figure 110). -

4 look up 4, A2, A), A4, and A5 for the dcelred angles of attack (Figure 109). - 5 compute -

+ A3 (R) + A4 (TI + 4 d

Numerical Example

Giver the following parameter., compute ~ A B S fdr a body-etrake corbination d

at Mach 1.2:

2 from Pigurel10:J2 - 0.0 -

4 from Figurelog -

Data Comparisons

I n F i g u r e l l l m e t h o d r e s u l t s are p l o t t e d a long v i t h those d a t a used i n formulat ing

t h e method. The power series y i e l d s a good approximation of xCPABS/d a c r o s s the

Mach ranbe. A lack of independent d a t a rt high ang les of a t t a c k makes f u r t h e r

, cocnparisons impossible a t t h i v tima.

It i r now a p p r o p r i a t e t o compare the c e n t a r of p ressure l o c a t i o n of t h e body 1

p lus s t r a k e conf igura t ion a5 ind ica ted by t e s t d a t a wi th t h a t determinedusing I previously de r ived methods. The fol lowing equat ion w i l l be used:

The methods used i n determining the va r ious components of the bas ic equat ion a r e

a s f o l l w s : I Component Source

C~~ Sect ion 5.1.1 (p. 39 f f )

A 'NBS Sect ion 5.3 .1 (PO 190 f f )

XcpB Sect ion 5.1.2 (PO 61 f f )

*CPABS Preceding a n a l y s i s

8alavant body paramterm are:

gtr- glrwterr are a8 contained in the aumarlcal example preceding.

Ure of t h e four methodo and app l i ca t ion of t he bas i c equation y i e l d r t h e

fo l lov lag r w u l t r a t .Hach 1.2:

Data Comparisons

Figure 112 compares the r e s u l t s of these empirical methods with a c t u a l

t e s t da ta f o r the body/strake configurat ion. Very good co r re l a t ion is shown

across the angle of a t t a c k range a t Mach 1 .2 .


Figure 107. General Curve Form, XCp hBS

ANGLE OF ATTACK-DPS.

ANGLE OF ATTACK-DEO.

Figurr 109. Polynomial Coefficient8 for Calculating XCp ABS

--.

ANGLE OF ATTACK-OEC.

--- ANGLE OF ATTACK-DEC.

?i(luta 109 (cont.). Poly11oPi.1 Coefficient* for Calbulating XCp ABS


Figure 109 (Cone.). Pelynodal Coefficient8 for Calculating

HACH NUMBER

Figure 110. J and K Values for Calculattng XCp ABS

MACH NUMBER

Xach 1.20

A Test Data - Power Series

X.ch 1.8.2.2

0 T e s t D8ta. Xach 1 . 8

v Test Data. Hash 2.2


Pigure 111Kont.). Comparism O f Teas Data And Herhod, XCp

RBS - d

5.3.3 Incremental Normal Force Due t o T a i l s

Summary

A method f a presented f o r p r ed i c t i ng ACN , t h e t o t a l increment caused BST

by t h e add i t i on of tails t o a body-strake conf igura t ion . Note t h a t AC N ~ s ~

inc ludes t he t o r c c s on t h e ta i ls a s w e l l as t h e carryover t o t h e body-strakes.

The anglemof a t t a c k range is 0 t o 180 .degrees and t h e Mach number range is

0.6 t o 2.2. Comparisons between pred ic ted and experimental r e s u l t s show

good agreement. This method , i s an ex tens ion of e x i s t i n g methods which a r e

accu ra t e a t ang l e s of a t t a c k approaching 0 and 180 degrees.

Background - The normal fo r ce on a body-strake-tai l conf igura t ion can be expressed

a s the sum of t he fo r ce s on t he i s o l a t e d components p lus in te r fe rence-

produced e f f e c t s and carryover between, t he var ious components. This

s ec t i on d e a l s with t he development of an empir ica l method which extends

t he present DATCOM method f o r p r ed i c t i ng t he increment i n normal fo r ce

due t o t he t a i l s o f , a body-strake-tail conf igura t ion . The extended method

covers the e n t i r e 0 t o 180 degree angle of a t t a c k range. l npu t s t o t he

aethod were obtained from DATCOM (Reference 17) a t t h r lower ang l e s and

experimental d a t a c o r r e l a t i o n s a t t he higher angles .

A t low to moderate angles of a t t a c k , say up t o 20 degrees, t he DATCOM

method extends t he ba s i c t h e o r e t i c a l procedures t o accouct f o r t he e f f e c t s

of separated flow i n the form of symmetric s teady v o r t i c e s . Sincs thb flow

p a t t e r n i n t he 0-180 degrees range usua l ly contairrs a s p e t r i c and/or unsteady

v o r t i c e s , n rnodlfication of t he DATCOM extension is inappropr ia te . In-

s t e ad , ,I new extension of the bas ic t h e o r e t i c a l procedures is des i r ed . The

new method w i l l p r ed i c t AC , which includes t he combined e f f e c t s of N~~~

interference and carryover. The nature of the inatrumentation and con-

figurations tested dictated the following formulation of the tail contri-

bution to n o r m 1 force: L

where C is determined by the method of Section 5.3.2. N~~

Method Developm~nt

A power series approach was used and in the usual way boundary conditions

were sought. First, values'of ACN were extract id from Wind tunnel data BST

on an Alr ~orce' body-strake-tail conf igtkation tested at angles of attack

between 0 and 180 degree8 and Mach numbers between 0.6 and 2.2 Uaing these

data as a guide, the values of ACN BST

at 0 and 180 degrees and a A C w ~ ~ ~ aa

at 90 degrees vere taken ss zero, The derivative aACNBST at 0 and 180 degrees aa

and the value of CN at 90 degrees were left aa free variables; viz., T (B)

and CN respectively. n/2'

Applying these boundary conditions to the power series expansion

a i n r ad i ans

Values of Ale A2 and A a r e p lo t t ed a s a func t ion of ang le of a t t a c k 3

i n Figures 113, 114 and 115.

Valuee of ACN and ACN can be determined using t h e methods of a0 On

Referencee 12 and 30. The general ized expression nuggeated i n Reference 30

used t o p red i c t t h e magnitude of ACN and ACN is a s follows: a0 an

ST - ' r ~ ( ~ ) + %('l')I ('N * Na i , 'a ref

where C N ~ , t h e t*?rmal f o r c e curve d o p e a t e i t h e r a - 0 o r a * n, can be a

determined using the method of DATCOM o r t h e RAS Data Sheete (Reference 27).

I n the care of ACNa , t h e va lues of %(B) and %(T) taken from Reference 30 0

can be determined from Figure 116. I n t he case of ACN , can be de ter - ", mined from Figure 116. Elovever, a t a - 1800 $(B) i r r a t equal t o 1.0 s i n c e

t h e r e w i l l be no upwuh due t o a forebody a t t h e "leading" edge of t h e t a i l .

Note that the s lope a t a - r w i l l be negktive.

Prom the experiment*lly der ived da t a , t h e va lue of ACN was found +/2

t o approxiar te 3.65 a t a11 M~ch numbera. Thia value app l i e s only t o t he

conf igura t ion t e s t ed , Assuming t h a t the value a t 90 degrees v a r i e s a s t n e I

r a t i o of planform areas , the following equation can be applied t o determine

velues cf AC f o r general configuraticne. Nll/2

Use of Method

The method fo r predict ing A C N ~ ( ~ ~ ) is applied i n the following way:

1 Determine CN using e i t h e r the method of DATCOM (Reference 17) or the - aT RAS Data Sheetr (Reference 27).

2 Calculate AC and ACN using Equation 41 and Figure 116. - -- - 3 Calculate ACN using Equation 42.

n/ 2 4 Using the r e s u l t s of s t ep r 2 and 3, Equation 40, a ~ d Figures -

113, 114 and 115 ca lcu la te ACN~(*S) between 0 and 180 degrees I. -

angle of a t t ack ,

Numer Pcal Example - Calculate ACN

BST between 0 and 180 degrees angle of a t t ack a t M =

0.6 fo r a configuration with the following c h a r a c t t r i ~ t i c s .

d - 3.667 in.

b s ingle panel - 1.867 in. exposed

S s ingle panel = 8.883 rq. in. T

% - 0.785 double panel ,

AT = 0.687

1 Using the RAS Data Sheets f o r X = 0.687 and AR - 0.785, the - slope of the t a i l normel force curve war determined t o be

1.173/rad.

\. . ., .

, 1'

I i

.. ;

i': >. .

i

/'. 1 4 .

>/'

2 Calculaee AC using Equation 41 and the r e e u l t e of s t e p 1. - Nao

A t a - 0 degree., u r i ag Figure 116 f o r dl. = 0.495, = 0 . 8

3 Cclcula te ACN ueing Equation 41 and the r e s u l t s of s t e p 1. - a u

A t a - 180 degrees, t h e r e w i l l be no upwash a t t he f i n "leading

edge" due t o a forebody; therefore . = 1.0; from Figure 116

ACN = - 3 . 3 l l r a d ; Sraf L a

4 Calcula te AC ueing t h e f s l lowing equation. -- N u / ~

5 Using Equation 40 the r e r u l t s of s t epe 2, 3 and 4 and Figures - 113, 114 and 115, c a l c u l a t e ACN between 0 and 18Q degrees

BST angle of a t t ack .

5 (Continued ) -

-. I Data Comparison

The results of the numerical example along with the, results of other

teat cases are compared against experimental data i n Pigure 117. Considering

the scatter in the data, agreement is good. Due to a 'lack of data on body-

#&rake-tail conf Qturations throu&hOu t the angle of attack range, independent

check8 of the method are no^ possible at t h i s t h e .

5 . 3 . 4 ' E f f e c t i v e Cen te r of P r e s s u r e f o r ~ n c r e m e n t a l Normal Force Due t o T a i l s

Suimary, /

A method i r presen ted to p r e d i c t XCp , t h e e f f e c t i v e c e r t e r of ABST

~ r 8 r e u r e o f t h e fnc rementa l normal f o r c e produce4 by add ing tails to a body-

s t r a k e c o n f i g u r a t i o n . The method is a p p l i c a b l e t o "plus" c o n f i g u r a t i o n a t

Mach numbers between 0.6 and 3.0 and a n g l e s of a t t a c k from 0 t o 180 degrees .

T h i s mec'hod h a s been a p p l i e d t o t h e c e n t e r o f p r e s s u r e c a l c u l a t i o n r f o r a

complete body-s t rake- ta i l c o n f i g u r a t i o n . Agreen~ent ,between p r e d i c t e d and

exper imental r e s u l t s were found t o be q u i t e good. In some c a s e s , i t was

found t h a t p r e d i c t i o n s could be improved by us'ing t h e Jorgensen t echn ique ,

f o r , p r e d i c t i n g C,NB up t o 40 d e g r e e s a n g l e of a t t a c k . U n t i l . enough comparisons

a r e a v a i l a b l e t o determine which method p rov ides b e t t e r r e s u l t s c o n s i s t e n t l y ,

i t is recommended t h a t bo th t h e C N ~ p r e d i c t i o n method of S e c t i o n 5.1.1 and

t h a t of Jorgensen (Reference 12) be used i n Rp BST c a l c u l a t i o n s up t o 40

degrees a n g l e of a t t a c k .

Background

Cur ren t methods f o r p r e d i c t i n g t h e e f f e c t i v e c e n t e r of p r e s s u r e , XCpABST,

of t h e increment i n normal f o r c e due t o t h e a d d i t i o n s of t a i l s t o a body-

s t r a k e c o n f i g u r a t i o n a r e no t a c c u r a t e over t h e e n t i r e 0 t o 180 degree a n g l e

of a t t a c k range. I n g e n e r a l , they a r e l i m i t e d t o a n g l e s of a t t a c k l e s a than

30 degrees . These methods r e q u i r e s e p a r a t e procedures t o c a l c u l a t e c e n t e r s

of ,.rLtisure f o r t h e t a i l i n ' t h e presence of t h e body, ca r ryover from t h e

t a i l t o t h e body and s t r a k e - t a i l i n t e r f e r e n c e . Using t h i s approach over t h e

e r l t i r e a n g l e o f a t t a c k range would r e q u i r e much more in fo rmat ion than was

a v a i l a b l e and would r e s u l t i n awkward and time c o n s u m i n g methods. In o r d e r

t o develop s imple , easy t o w e methods f o r p re l iminary des ign purposes , a '

method is presented for calculating a composite center of pressure for the

total increment in normal force due to tne addition of tails.

Method Development

An analytic approach to method development was ruled out due to the

complicated nature of the flov field. A power series approach to method

development was selected and, In the usual way, boundary conditions were

sought. Available experimental data were of little use in determining ooundary

conditions. The only data available were total configuration pitching moment

and normal force coefficients for body-strake-tail and body-strake configura-

tions. Applying these data to the following equation yielded highly questionable

results.

At angles of attack greater than 90 degrees, calculated centers of pressure

were off the body. Thie can be attributed to the effect of tail downwash on .. the etrakes. Tail downwash will lower the normal force on the strskes and

tend to move the strake X aft. This results in a much larger change in CP

moment due to the aJ4+ion of tails than the change in normal force would

tend to indicate. Keeping this in mind, other sources of boundary conditions

were sought.

At a - O degrees, the effective center of pressure of the incremental force due to the addition of tails can be approximated by summing the moments

about the tail leading edge at the root.

XcP ABST

C~

s he important incremental forcea a r e taken t o be t he forcd on t he tail

i n t he presence of t he body and t h e fo r ce cn body i n t he presence of t he t a i l .

Teme accounting f o r t he effect . of e t r ake v o r t i c e s on t he t o i l are not In-

cluded, 8ince a t o * 0 degrees atrake v o r t i c e s w i l l be weak o r non-existent.

Values of x c ~ X c ~ %@)* K ~ ( ~ ) ' T@L, and B ( T ) can be found in References 4 I'

'R C~

and 30. However, f o r the ' sake of cornpleteaess, they a r e presented aga in

can again b e used. s ( B ) ' s h o u l d be equal t o 1 .0 s i n c e t he r e w i l l be no

upwaeh st the t a i l t r a i l i n g edge due to t he preeence of a forebody. Values

x c ~ x c ~ of T(B) can be taken from Figures 119 and 120. Values f o r JT) are

C~ C~

,presented i n Reference 30, Again f o r t he sake of completenees these, va lues

a r e presented here i n Figures 123 and 124. A t a - 90 degree8 t h e r e w i l l be

no in te r r ' e renre between t h e t a i l e end s t r a k e s , Then t he e f f e c t i v e c e n t e r of

p ressure can be assumed a t t he cen t ro id of t he f i n pianform area . Thie

assumption i s v a l i d so long as t h e car ryover from the tai ls t o the body Is

amall. Sect ion 5.2.2 dea l ing with I h80 shorn t h a t t he car ryover in B (TI rmall.

Applying the above boundary condi t ions t o t h e fol lowing power s e r i e s

expaneion

y ie lded

which can be rewritten as:

where :

( a in radians )

Values of Ao. A1,' and A are plotted in ~ i p r e s 125. 126 and 127. 2

Use of Method

A general deecription presenting the details of how .to apply this method

will be presented in this section. This will be followed by a numerical

example in ,which this method is applied in conjunct?on with the other methods

needed to calculate the XCp of a'complete body-etrake-tail configuration.,

X 1 Calculate CPO - -

a Uee Figure 118 to d-termine values of - '(T(B) and at

the appropriate -.als,i of d/,i.

b Depending upon the H.ch number, use either Figure 119 or 120 - X

to determine CP T (~1' C~

c Depending upon' the Mach number, use either Figure 121 or 122 - X

to determine CP B[T)

C~

d Apply the r e e u l t e of s t e p s a, b and c t o Equation 43. The - ca lcu la ted cen te r of preseure i e measured from the leading

edge t o t he f i n roo t chord.

X 2 Calcula te CPn - -

a Use Figure 118 t o determine - KB(T) and assume %(B) = l*"*

b Depending upon the Mach number, uee e i t h e r Figure 119 o r 120 - X t o determine CPTO) .

c~

c Depending'upon the Mach number use e i t h e r Figure 123 o r 124 t o d

d Apply the r e s u l t 8 of Steps a, b and c to, Equation 43. The , - '

ca lcu la ted cen te r of prsesure is measured from t h e t r a i l i n g

edge of t he f i i ~ roo t chord.

e Using the r e s u l t s of s t e p d determine t h e center of pressure - a s measured from the leading edge of the f i n root chord.

3 Calculate the cent ro id of the f i n plnnfo& a rea a s measured from - the leading edge of the f i n root chord.

4 Apply the r e s 3 l t s of s t e p s 1, 2 and 3 t o Equation 44 t o determine - X

the of ACN f o r angles of a t t a c k between 0 and 180 degrees. BST

c~ Numerical Example

Calcula te the cen t e r of preseure f o r the following body-strako-tail

conf igura t ion a t M = 0.6.

Body:

e a~ tangent - * 14.5 d - 3.667 in. - - 2.5 ( d ogive 5

Strakes:

C% = 14.33 in.

b - = 0.40 in. 2

Tails:

bT - 1.867 in.

X a Calculate CP - 0 - ,.

d i. Using Figure 118 for - - 0.495, 8

%(B) - K~ (T

= 0.8

11. From Figure 119 for ATaE = 0' and X - 0.687;

d iii. From Figure 121 for A = 0.687, 2 = 0.495 and no

, afterbody:

l v . Applying the results of eteps i through iii to

Equation 43 y i e l d s :

50) - 1.0 in the absence of a forebody.

11. Prom Figure 119 for A - 0.687 and facing forward)

f ii.

iv .

Prom Figure 123 for 4 - 0.495, A - (fee.. Fin trailins edge forward):

Y

"La -, O0

0.687 and itE 0'

A p p l i , the re&lt. of mtep i through iii to Equation

43 yields t

-..I = 0.142 (measured from T,E,) OR

d Apply the reeul t s of s t ' e p a, b and c t o Equation 44 - for angles of attack between 0 and 180 degrees.

u r i q the method of Section 5 . 3 . 3 (p. 220 f f )

Calculate AC and X N~~ CpABS

using the methods of Sections 5.3.1

.Ird 5.3.2, respectively. (p. 190 ff)

X 4 Calculate C and CP ueing the method8 of Sectlona 5.1.1 - I % B -

d I I

(P. 39 f f ) acd 5.1.2 (p. 61 f f ) , respectively

5 Calculate the centers of pressure bekween 0 and 180 degrees for the - complete body-strake-tail configuration using the following equation:

except a t a - 0 and 180 degrees

x c ~ ' CM a when [-I = -

cm ABST

t Data Comparisons

ABST

The results of the numerical example are compared with experimental data

at Hach 0.8 in Figure 128. As can be seen, agreement is q u i t e good. The

rerulccr obtained using Jorgensen's C N ~ predictiooe up to 40 degrees are

also preeanted. The r e s u l t 8 of further check cases at other Mach numbera

are show i n Figures 129, 130, and 131. As noted i n Section 5.1.1,

Jorgensen'e method is rccomended for predicting C N ~ up to angles of

attack of 40 degrees.

Reference 30

BODY DIAMETER d - -- SPAN 8

Figure 118. %(B) and Ratios

0.6 T-- Extrapolation

0.4

0. ?

0

0.6

0 .4

x c ~ T - 0.2

0

0.6

0.4

0.2

/ -- I

h 0 1 2 3 4 5 6 7 8

EFFECTIVE ASPECT RATIC BAR

(a) No Leadfng-Wue Sveep (b) No ~nidchord Sweep (c) No Trailing-Edge !beep

ll#ure 119. T a i l Alom Center of Pressure a t Su!amnic Speeds

EFFECTIVE ASPECT RAT 10. PAR (a) No 1,~sdhrg-F.dgc %wcp (h) No Mtdchord Svrep ( c ) No l't-at1 t n ~ - E d ~ t '

PCgttre 1,20. Tail Alone Center of Pressure a t ~&ermonfc Spcada

Reference 30

Figure 121. Curve* for Determining X /CR for Subaonic Speeds , "B(T)

(Zero Trailing Edge Sweep)

Reference 30

EFFECTIVE ASPECT RATIO BAR

Figure 122. Curves for Determining XCp /CR for No-Af terbody B (TI

at Supersonic Spaedr

Reference 30

I I a/p - 0,0.2,%4,0.6

I

I

I

0 1 2 3 4 5 6 7 8 GPPECI'NE ASPECT RATIO, BAR

Figure 123. Curves for Determining XCp /CR for Subsonic Speeds 0 0)

(Zero Leading Edge Sweep)

Figure 124. Curve8 for Determining tp /CR with Afterbodies

B (T) at Superronic Spaedr


Figure 128. Comparison Between Predicted And Experimental Results, xCPBsT . 14 = 3.6 -.-

ANGLE -0P ATTACK-DPC . Figure 129. ~ompa&on Between Predicted And Experimental Results, X

CPBS.,! - 0.8s - d

5.4 Fdy-Wing-Tail Conf fgura t ions

5 . 4 . 1 Incremental Normal Force Due t o Wings

S u m r y

A method is presen ted t o p r e d i c t As , t h e t o t a l increment i n normal BW

f o r c e due t o t h e a d d i t i o n of wings t o a body. The method is a p p l i c a b l e

t o Mach numbers between 0.6 and 3.0 and a n g l e s of a t t a c k from 0 t o 30

degrees . Comparisons between p r e d i c t e d result!, and exper imental d a t a show

good agreement f o r a l l c a s e s except Xach numbers L ~ R R than 1.0. The maximum

d i f f e r e n c e between p r e d i c t e d and exper imental v a l u e s f o r t h e s z subsonic '

c a s e s o c c u r s a t an a n g l e of a t t a c k of 30 d e g r e e s cad can amount t o an

u n d e r p r e d f c t i o n o f between 30 and 40 p e r c e n t . A d18cusaion of soee poselb1,e

s o u r c e s of t h e d i sc repancy is presen ted in ' connec t . i zn wi th t h e compariwms

between test and p r e d i c t e d va luee .

9ec karound

.Addi t ion of wings t o a body w i l l produce a n i n c r e a s e i n normal io rke .

T h i s i n c r e a s e d i f f e r s from t h e normal f o r c e produced on t h e i s o l a t e d wing

m d e r i d e n t i c e 1 f r e e str,eam c o n d i t i o n e . The d i f i e r e n c e is a t t r i b u t a b l e

t o anrtuaL i n t e r f e r e n c e s between c o n f i g u r a t i o n components. A t low a n g l e s

of a t t a c k (ac6"), t h e i n t e r f e r e n c e e f f e c t s a r e due l a r g e l y t o body upwgsh

on t h e wings, normal f o r c e carry-ovdr from t h e wings t o t h e body and down-

wash imposed on t h e body a f t of t h e wings due t o t r a i l i n g wing v o r t i c e s .

As a n g l e of a t t a c k is inc reased beyond 6 degreeo, t h e body c r o s s f l o w bou3dary

l a y e r b e g i n s t o o e p a r a t e and r o l l up i n t o symmetrically d i sposed v o r t i c e r

on e i t h e r s i d e of t h e body. Downwash from t h e s e v o r t i c e s ha8 an a d d i t i o n a l

e f f e c t on wing load ing . Body v o r t i c e o grow i n r im and s t r e n g t h w i t h

i n c r e a s e s i n a n g l e of a t t a c k ; t h e r e f o r e , t h e i r i n f l u e n c e v a r i e s w i t h a n g l e

o f i t t ? .~ck . A t i l l l ~ l e ~ of t r t t e c k g r e a t e r t han 30 d e g r e e s , t h e v o r t e x wake

w i l l bccomr asymmt*tric J u e t o t h e a l t e r n a t e shedd ing and growth o f

atidf t t ona l v o r t I c e s . An nsymmetric body v o r t e x wake w i l l , a l t e r tllc downwash

which each wing c x p e r l t m - e s . T h i s i n t u r n w i l l a l t e r t h e l o a d i n g on each

wlng indue fng .I w n f l g u r a t Ion r o l l i n g moment. T h i s problem is more a c c u t e

i n tht* silbson i c and t r a n s o n i c Mach regimes .

Kt? t hod Deve>2>pi-~yL

A method is rcqul rec l t o p r e d i c t t h e increment i n norm;~l f o r c e due t o

th=* a d d i t i o n o f wings t o a body. ACNBW. The method is t o be a p p l i c a b l e t >

M x h numbers between 0 . 5 and 3.0 and a n g l e s o f a t t a c k t o 30 o r 40 d e g r e e s .

The method must accoun t f o r wing non- l i nea r normal f o r c e c h a r ~ c t e ~ i s t i c s and

mutual component i ~ t e r f r x e n c e s .

The e x i s . :rg method o f DATCOM (Re fe rence 17 ) p r e d i c t s i s o l a t e d 4%

normal f o r c e r~s a f u n c t i o n o f a n g l e o f c t t n c k , i n c l u d i n g n o n - l i n e a r e f f e c t s .

Wing normiil . o r r e p r e d i c t i o n s are c o r r e c t e d f o r i n t e r f e r e n c e e f f e c t s u s i n g

t h e s l e n d e r body i n t e r f t * r e n c e f a c t o r s o f Re fe rence 30. Body v o r t e x e f f e c t s

on t h e wings a r e p r e d i c t e d s e p a r a t e l y and added t o t h e s e r e s u l t s . The

p r o c e d u r ~ f o r p r e d i c t i n g body v o r t e x e f f e c t s r e q u i r e s t h e p r e d i c t i o n of v o r t e x

l o c a t i o n and s t r e n g t h i n o r d e r t o d e t e r m i n e v o r t e x i n t e r f e r e n c e f i l r t n r s .

Body v o r t e x i n l e r f e r e n c e f a c t o r s a r e a p p l i e d t o wing 1 i n e a r normal f o r c e

characteristics o n l y . I n F i g u r e 132 t h e method o f DATCOM has been a p p l i e d

t o a body wtnu conf i g u r n t ion and t h e r e s u l t s compared wt t h e x p e r i m e n t a l

d i ~ t a o f R e f e r e n c e 30. I'he compar ison between p r e d i c t e d and e x p e r i m r n t a l

r e s u l t s is q u i t e good up t o 20 d e g r e e s a n g l e of a t t a c k . However, t o

ex t end t h e p r e d i c t i o n s p a s t 20 d e g r e e s r e q u i r e s e x t r n p o l a t i o n . For t h e

comparison of F i g ~ ~ r e 132 t h e p r e d i c t i o n s were extended t o 25 d e g r e e s . The

compar ison shows t h a t p a s t 20 degre ; e s , p r e d i c t i o n s and e x p e r i m e n t a l d a t a

d i v e r g e . Due LO t h i s a n g l e of a t t a c k l i m i t a t i o n ani! d i f f i c u l t i e a e c c o u n t e r e d

w i t h t h e body v o r t e x i n t e r f e r e n c e p r e d i c t i o n methods, a new w t h o d

was d twe lop td .

The method of t h i s s e c t i o n p r e d i c t s i s o l a t e d wing norm81 f o r c e co-

e f f i c i e n t s a s a func:ion of d n g l e o f a t t a c k and t h e n c o r r e c t s f o r i n t e r f e r e n c e

e f f e c t s . U t i l i z i n g t h e concep t of t h e i n t e r f e r e n c e f a c t o r s a c c o r d i n g t o

Refe rences 1 7 and 30, t h e i n c r e m e n t a l normal f o r c e due t o t h e a d d i t i o n o f

a wing t o a body 19:

where i (a) r e p r e s e n t s :he i s o l . a t e d s u r f a c e c o e f f i c i e n t s . N w %(B) and B (W)

a r e interference f a c t o r s . Methods f o r p r e d i c t i n g CN~(II) and R a s e W ( B )

p r e s e n t e d i n S e c t i o n s 5.1.4 and 5.2.1, r e s p e c t i v e l y . Empi r i ca l i n p u t s t o

t h e s e methods were developed u s i n g t h e d a t a of Re re rence 13. H W(B) is an

L n t e ~ f c r e n c c f a c t o r which wtitw a p p l i e d t o i s o l a t e d panel d a t a p r e d i c t s t h e

norm.ll f o r c e on he wing i n t h e p r e s e n c e of t h e body. RW(B) e m p i r i c a l l y

; ~ c c o u n t s f o r body upwash and body v o r t e x downwtish on t h e wing and can b e '

p r e d i c t e d a s n f u n c t i o n of M , d / , s , X anci tr. R r e p l a c e s t h e \ and W ( R ) (8;

body v o r t e x term in t h e method of Refe rence 17. The normal f o r c e on t h e

wing in t h e prt*senc-r of t h e body is b e l i e v e d t o be t h e d o m i n ~ ~ t i n g f a c t o r

i n , t h v A C N ~ ~ term. T h e r e f o r e f o r t h e pu rposes of t h i s method, t h e normal

c . ~ r r y - o v e r from t t ~ c wing t o t h e body can be p r e d i c t e d w i t h suf f i c i e i i t accu racy

u s i n g t h e ca r ry -ove r f a c t o r K B(W)

o f Re fe rence 30. For t h e s a k e o f comple te-

I ~ C S S , v a l u e s o f K B (W)

a r c p r e s e n t e d a g a i n i n F i g u r e 133.

Use of Hcthod -- ----- A genera l d e s c r i p t i o n of how t o app ly t h e method w i l l be presen ted

i n t h i s s e c t i o n . T h l ~ w i l l bc fol lowed by a numerical example dcmondt rn t ing '

1 C a l c u l a t e i s q l a t r d wjng norm11 f o r c e c o e f f i c i r ~ ~ t a a 8 (I - f u n c t i o n of kng le of n t t n r k o s t n g t h e mt%thotl of S e c t i o n 5.1.4.

2 ' C s l c u l a t c t h e i n t e r f e r e n c e f a c t o r \(B) us ing t h e m c t l w d - of Scst i ,on 5.2.1. (p. ? 4 3 i f ) .

3 U s e Figure 133 t o de te rmine K a t t h e npprop , r i :~ tc v ~ l u r of - B ( W )

d j s .

4 Applv t h e r e s u l t s of S t e p s 1-3 t o Equ i~ t ion b 5 .

Numt-r i c a l Examn-l~ - ------ The method f o r p r e d i c t i n g ACN ((I) w i l l be a p p l i r d t o n conf i ) ;ura t ion 3.W

v i t h t h e fo l lowing c \ ~ a r a c t e r i a t i c s a t M - 1 . 1 .

Body:

t l d - 10.0

rNji - 3.0 t a n g c n t - o l i v e

d - 3.75 in .

Wing:

The s t e p s a r c a a fo l lows :

_1 C a l r u l a t i o n of C w i n g t he method of S e c t i o n 5.1.4 (P. 91 f f ) . Nu

C ( 0 ) - 2.93/r.d from R.A.S . Data Sheeta (bference 27) N~

(1

* b i n g Transonic method

4 Apply t11e r e s u l t s of S teps 1-3 t o Equation 45. -

Data Comparisons - The r e s u l t s of t he numerical example a r e compared d i t h experimental

da t a i n Figure 134. See ~ i ~ i r e 1 3 5 ' f o r a sketch of t he conf igura t ion .

Further comparisons between pred ic ted r e s u l t s and experimental data a r e

presented i n Figures 136 through 139. These comparisons cover a range of

Mach ?umbers and conf igura t ions . The configurg't ions of i n t e r e s t vary body

length , r e l a t i v e wing s i z e and wing pJanform. In a l l cases agreement between

predicted and experimental d a t a is q u i t e good except a t eubsonic Mach

numberd. See Figure 139. For t he conf igura t ions of Figure 139, t he pre-

d i c t ed and experimental r e s u l t s begin t o d iverge r ap id ly between 22 and

30 degrees angle of a t t a ck . The maxlmum d i f f e r ences occur a t 30 degrees

where the predicted r e s u l t s a r e between 30 and 40 percent under t he

experimental values. A t t h i s time, t h e source of the d i f f e r ence cannot

be determined. However, one aspec t of t he proposed method must be considered

a s suspect , namely the use of K from Reference 30 which s t r i c t l y speaking B (W)

a p p l i e s a t angles of a t t a c k near zero only. Since t he wings were not

instrumented i n t he t e s t s which provided t h e b a s i s f o r t he cu r r en t s tudy,

t he v a r i a t i o n i n K B (W)

with a cannot be evaluated. Therefore, R W ( B ) , which

is t h e s a w a s R T(B)

(Sec t ion 5 .2 .1) . a c c o u n t s f o r t h e e f f e c t s of a , b u t

does not . I n o r d e r t o e x p l o r e t h e s e n s i t i v t c y of t h e r e s u l t t o K B (W) '

v a l u e s of K were chosen such t h a t agreement v i t h t h e t e s t d a t a was B (W)

achieved. The r e q u i r e d v a l u e is t v i c e t h e magnitude am expected f o r t h e

p a r t i c u l a r hor'y d iamete r t o span r a t i o . For example, R ~ ( ~ )

f o r c o n f i g u r a t i o n

1 was set equa l t o 1.46 ( t h e maximum v a l u e i n d i c a t e d by t h e d a t a of Reference

13 f o r n f i n w i t h d/s = 0.5) and t h e v a l u e of K which would f o r c e B (W)

,matching was c a l c u l a t e d . The v a l u e c a l c u l a t e d was approximately 1 . 6 o r

twice t h e v a l u c of 0.8 predic , ted i n F igure 133. A s a r e s u l t , a q u e s t i o n

can be r a i s e d concern ing the accuracy of t h e tes t . d a t a . F u r t h e r s y s t e m a t i c

d a t a i r n e c e s s a r y t o de te rmine i f the d i f f e r e n c e s observed i n F i g u r e

139 a r e due t o i n a c c u r a c i e s i n t h e exper imen ta l d a t a or i n t h e g r e d i c t i o ~

m e t hod.

0 Experimental (Ref. 3 4 ) - DATCOU (Ref. 17:

10 20

ANGLE OF ArTACX-DEC.

Reference 39

BODY DIAMETER d * - SPAN 8

Figure 133. KBCW) Ratio at Zero Angle of Attack

(I) \ a


, , 0 E x p e r i m e n t a l (Ref. 20)

- Predicted

6 . At:

X = 0.4

N~~ AR - 0.514

0 10 20 ' ANGLE OF ATTACK-DEG.

F i g u r e 1 3 7 . Cornp;~rison B e t w e e n E x p a r h e n t a l And P r e d i c t e d R e s u l t s , *"BW

, F1-3.08

0 Expertmentnl (Rcf. 35) - Prrdtctcd

H - 1.9

ANGLE OF, ATTACK-DEG.

Figure 138. Comparison Betveen Experimental And Predicted Results,

bCNBh' ' * I rn9

5 . 4 . 2 Effective Center of Pressure for Incrqmentzl P o m l Force Due to Wings

Summary

A method to predict the effective center of pressure, x ~ ~ i m w * Oi the

incremental normal force, ACNw. is presented. AC includes normal force N~~

on the wing in tte presence of the body plus any carry-ove- from the wing

to the body. The method is applicable to Mach numbers from 0.6 to 3.0

and angles of attack from 0 to 30 degrees for body-wing configurations.

Comparisons between predictions and experimental data have shown good

agreement.

Background

The addition of wings to a body produces an incremental normal force,

ACNBW. This incremental normel force includes the normal force on the wing

in the presence of the body plus any carry-over from the wing to the body.

See Section 5.4.1. When attempting to predict wing-body.configuration aero-

dynamic stability characteristics, it is necessary to determine the effective

center of pressure, of ' A C ~ For prelimibry design purposes, it BW

ie desired that the method for predicting this center of pressure be easy

to use. The current met'tod of Reference 17 is awkward to employ. There-

fore e more elementary, easy to use method'will be preeented in this

eec tion.

Method Development

Development of the method began with an analysis of experimental data

of References 20 and 34, consisting of normal force and pitching moment

coefficients for isolated bodies and body-wing configurations. Experimental

value8 of t h e XcpABW were determined uaing t h e fo l lawing equat ion:

C ~ B w 'CPBW _ c x c ~ B XcpABW -- d

d AC.. N~ d

where cen t e r o f , p r e s s u r e is measured i n diameters from the nose. The

r e s u l t s showed t h a t X /d remained e e s e n t i a l l y cons tan t Tor angles of @ABW

a t t a c k between 0 and 30 degrees. Therefore, t he method w i l l r e l y on ,

pred i c t i ng XCp a t a - 0 degrees and l e t t i n g i t , remain cons tan t between 0

and 30 degrzes.

To make t he method independent of forebody length , t h e procedure

de f lne s t he cen t e r of p ressure l oca t i on a s a percentage of t he wing roo t

chord measured from the roo t chord lead ing edge. Based on t he d i s cus s ions

of Reference 30, iPABV /cR a t a- 0. can be expressed as:

where CN terms cance l and IC, and XCP /Cg terms, a~ B W )

as derived from s lender body theory, a r e presented i n Figures 140 throbbh,

Use of Method 'cpABw ,

To i l l u s t r a t e f u l l y t he use of t he method f o r p r e d i c t i n g , CP , a

general description of t he procedure is presented, followed by a

s t e p by s t e p numerical example.

1 Using F igure 140, determine t h e va lue8 o f - %(b) and B(W) Zor t h e d / s of i n t e r e s t .

2 Depending upon t h e Mach nunbhr, u s e e i t h e r F igure 141, 142 -

3 Depending upon t h e Mach number, u s e e i t h e r ~ i g u r e 143 o r 144 - t o determine x ~ p a (U) f o r wings w i t h a f t e r b o d i e s .

n

A , A C P ~ ~ ~ 4 Using Equat ion 47, c a l c u l a t e "ABW a t a = 0 degrees . - ..

remains f i x e d f o r a n g l e s of a t t a c k between 0 and 30 degrees .

5 To express -xcpABU i n terms of d iamete r s from t h e nose u s e t h e

C~

fo l lowing equation.

Numerical Example

X C a l c u l a t e CPABW a t M - 0.85 f o r a body-wing c o n f i g u r a t i o n wi th t h e ,

d fol lowing c h a r a c t e r i s t i c s .

= 1 0 d = 3.75 it* 'LE = 16.75 in.

X = O AR = 2.0 d / s = 0.5 CR = 3.75 in.

1 From Figure 140, f o r d/s = 0 .5 - %(B) -

X 2 Since M = 0.'85, use Figure 141 to determine -3 for A = 0. -

C~

X c ~ 3 Since M = 0.85, use Figure 143 to determine B(W) for X = 0 and - m

4 Apply the results of Steps 1 - 3 to Equation 47. , -

5 Express ACPABW in terms of diameter? from tho noas. - C~

Data Comparisons . .

A sketch of the configuration' used in the numerical example is presented

X in Fip:!re 145. The values of CPABW/CR calculated in the numerical example

plus rewlts for the other configurations of Figure 145 are compared with

experimental data (Reference 34) in Figure 146. Further comparisons are

presented in Figure 147 for the tame configurations at M = 1.1. Figures

148 and 149 compare predictions with experimental (Reference 35)

centers of pressure. The predicted values of center of pressure for the

body-wing com5ination requires: the method of Section 5.1.1 for the body

normal force coefficient (CNB), the method of Section 5.1.2 for the

,'

cen t e r of p ressure X C P ~ Of % , t he method of Sect ion 5.4.1 f o r t he

incremental normal fo r ce c o e f f i c i e n t due t o t he add i t i on of a wing t o a \

body ( A C N ~ ~ ) , and f i n a l l y t h e method described i n t h i s s ec t i on f o r t he

effective cen t e r of prneeure XCPABW of A$BW- The components a r e cam-

bined a s fol lows t o ob t a in t h e t o t a l conf igura t ion cen t e r of p ressure .

r>)+%J+) %P I

d C + bCN N~ BW

'.

Reference .30

0 0.2 0 .4 0 . 6 1.0 BODY DIAMETER fi

SPAN 8

I Reference 30 I

Sffective h p e c t Ratio, BAR

( n ) No Leading-Edge Sweep 0 ) No Hid-Chord Svesp (c) No Trailing-Edge Sveep

Figure 161. Ufng Alone, ccnter 'o f ' Reswre At Subeonic Speeds

Reference 30

' Effective Aspect Ratio, BAR

Figure 14?. Curv~a for Determining XCp /CR a t Subronic Speeds EOJ)

Figure I/+&. Cudver for Determining XCp /c,, vith Afterbody at Supersonic Speeds B Dr)

CR-4.333d AR-1.231 - Predicted CN/d - 2.5 d/s - 0.273

0 5 10 15 20 2 5


Figure 148. Comparison Between Predictlone And Experimentnl Data.

d

L/d rr 10.333 1-0 C,, - 4.333d . 1-23, 0 Experimental (Ref. 35)

Figure 149. Comparison Between Predictiona And Experimental Data, XCp ,

ABW - n

5.4.3 T a i l Incremental Normal Force Due t o Wing-Vortex Interference

Surmnary

A method is presented f o r p r ed i c t i ng A C u W , the incremental normal

fo r ce produced oir, a t a i l .due t o wing-vortex , in te r fe rence . The method

p r e d i c t s a vort.ex induced ang l e of a t t a c k a t t he t a i l , E, which can be used

i n conjunct ion with i so l a t ed t a i l normal fo r ce d a t a t o de f ine LCNTWV. The

method accounts f o r v a r i a t i o n s i n wing- ta i l spacing f o r angles of a t t a c k t o

30 degrees i n t h e t ransonic regime. Supersonic capabilities, however, a r e

l imi ted t o 24 degrees angle of a t t dck . I n s u f f i c i e n t d a t a were a v a i l a b l e f o r

c o r r e l a t i o n a t angles g r ea t e r than 24 degrees. Data a v a i l a b l e f o r cu r r e l a -

t i o n i n both Mach number regimes represented l imi ted v a r i a t i o n s of wing and

t a i l geometries. However, comparisons between pred ic ted and experimental

r e s u l t s f o r geometries not used

a p p l i c a b i l i t y over a wide range

r e s u l t s have been obtained i n a 1

Beckground

i n t he c o r r e l a t i o n have demonstrated '

of wing and t a i l geometries. Reasonable

1 check cases.,

T a i l loads f o r body-wing-tail conf igura t ions d i f f e r from those of

body-tai l conf igura t ions . The d i f l e ronce is due t o wing-tai l i n t e r f r r c n c e

caused by v o r t i c e s t rc -? l ing a f t in the f r e e 'stream d i r e c t i o n from a I f f t i n g

wing. According t o the Kutta-Joukowski r e l a t i o n s h i p , t he s t r eng th of these

t r a i l i n g v o r t i c e ~ is r e l a t ed t o wing l i f t . A s t he v o r t i c e s stream a f t they

a r e displaced l a t e r a l l y and v e r t i c a l l y by body crossf low and mutual vor tex

i n t e r ac t i ons . These t r a i l i n g v o r t i c e s a i t e r the f lowf ie ld encountered by a

t a i l su r f ace and t he re fo re change t he t a i l l cad ing . Assuming p o t e n t i a l

v o r t i c e s (Vt a 3 / r ) , vortex in f luence on the t a i l diminishes with increased

s epa ra t i on d i s t ance between the vortex core and the t a i l sur face . To

develop a method f o r p r ed i c t i ng t h e incremental normal fo r ce (ACNTWV)on a

t a i l due t o wing v o r t i c e s it w i l l be necessary LO account f o r vor tex

s t r eng ths and v a r i a t i o n s i n vo r t ex - t a i l separa t ion d is tances .

Met hod eve lop'men t

'Values of A C N ~ ~ were ex t rac ted from experimental da t a using t he

following expression :

T a i l i n presence of T a i l i n presence of body and wing + body + (48)

carryover , carryover \

I A c ~ ~ ( C N B ~ - C N ~ ~ ) ' - N - CNB)

The q u a n t i t i e s C N ~ ~ ~ , CNBw, C N ~ ~ and C N ~ represen t main balance d a t a from

conf igura t ion build-up t d s t s . The assumption was made t h a t the t o t a l

increment i n normal f o r c e , obtained using Equation 48, is appl ied t o t he

t a i l panels only. According t o Reference 30, t he por t ion of t he incremental

normal fo r ce c a r r i e d over t o t he body w i l l genera l ly be a small f r a c t i o n of

t he t o t a l incremen:.

Data which could be appl ied t o Equatibn 48 were l imi ted . Most of t h e

d a t a were from a t ransonic body-wing-tail build-up test (Reference 34)

f o r angles of a t t a c k t o 30 degrees. Wings and t a i l s t e s t e d were l im i t ed

t o aepect r a t i o 2.0 and t ape r ra t ' io 0; howevcr, winp d / s and wing-tai l

a x i a l spacing were sys temat ica l ly var ied a s i l l u s t r a t e d i n Figure 150.

Supersonic d a t a were not ava i l ab l e f o r t he same conf igura t ion t e s t e d

t r anson i ca l l y . Sce Reference 20 f o r a de sc r ip t i on of t he supersonic t e s t

conf igura t ions . Supersonic da t a were l im i t ec -0 22 degrees angle a € a t t a c k .

To analyze t he r e s u l t s obtained by applying ransonic t e s t da t a t o

Equation 48, ACNTw was equated t o t he normal force produced by an i s o l a t e d

t a i l a t an angle a t t a c k , a. Therefore, a is analogous t o t he e f f e c t i v e

t a i l angle of a t t a c k , E , induced by t he presence of a t r a i l i n g wing vortex.

Ur5n.s t he values of A C N ~ ext rac ted from the t ransonic t e s t da t a and

i s o l a t e d f i n da t a , v d u e s of c were determined.

The method presented i n t h i s s ec t ion was designed t o predic t c. This

angle can be used i n conjsnct ion with i s o l a t e d f i n da t a t o determine bdN, TWY'

Important parameters which must be considered when attemptfng t o p red i c t €

a r e vo r t ex - t a i l separa t ion d i s t ance and vortex s t r eng th .

Var tcx- ta i l cepara t ion d i s t ance is a funct ion of configurat ion angle of

a t t a c k and wing-taf l a x i a l separa t ion d i s t ance (See Figure 151). A wing

vor tex sheds a t a l a t e r a l pos i t i on approximated by = - nb, t he pos: t i o n 4

predicted by' s lender body theory f o r low aspec t r a t i o f i n s . According t o

Reference 30 the v o r t i c e s t r a i l a f t from the wing t r a i l i n g edge a t an angle

of a t t a c k equal t o t h e f r e e stream angle of a t t ack . The v e r t i c a l d i s t ance

h , repara t ing the vor tex center and t a l l is ' defined a t t he point where t h e

vor tex breaks t h e plane of t h e t a i l l ead ing edge a t a l a t e r a l pos i t i on \. In, the case where yw is g r e a t e r than the t o i l semispen. t h i s poni t ion is

defined a s t he point a t which t h e vor tex core i n t e r s e c t s a plane perpendicular

t o t he body center l i n e and passing through t h e i n t e r s e c t i o n of t he t a i l

leading edge and the t i p chord. The v e r t i c a l d i s t ance separa t ing the vor tex

core and t a i l is e-rpreseed non-dimensionally as:

& - & t an a d d

where I, i s ' d s f i n e d as the axial d i s t ance between the wing t r a i l i n g edge

and the t a i l leading edge.

According t o t he Kutta-Joukwski r e l a t i onsh ip , vortex s t r eng th is r e l a t e d

t o l i f t . Therefore, normal fo rce on the wing i n the presence of t h e body,

i&W(B), vas u t i l i z e d a s t he measure of vor tex s t r eng th . Variat ions i n

vortex s t r eng th due to Mach number, plznform and angle of a t t ack can be

r e f l e c t e d by CN W(B) ' a )LIZ The measured v a l u e s of c c o r r e l a t e d w e l l w i t h ( C N ~ ( ~ ) 9 (F igure

152a). T h i s term i n c o r p o r a t e s t h e major parameters r e l a t i n g v o r t e x

s t r e n g t h and v o r t e x - t a i l s e p a r a t i o n d i s t a n c e t o v o r t e x i n f l u e n c e on t h e t a i l .

For a n g l e s of a t t a c k up t o 1 6 degrees , c was found t o va ry l f n e a r l y w i t h t h e

d 112 p a r a a e t e r ( - ) h CN~(~). A t a n g l e s of a t t a c k of 16 t o 24 d e g r e e s , v a l u e s

of t/q6 were found t o c o r r e l a t e as a f u n c t i o n of a n g l e of a t t a c k and t o be . independent of Mach number. However, f o r a n g l e s of a t t a z k g r e a t e r than 24

degrees , € 1 ~ ~ 6 became a s t r o n g f u r ~ c t i o n o f Mach number i n t h e t r a n s o n i c range.

Sea F igure 152b. Note t h a t €16 i n F igure 152b correcponds t o t h e v o r t e x

inducd a n g l e of a r t a c k a t a - 1 6 degrees .

There were i n s u f f i c i e n t d a t a e v a i l a b i e t o determine what caused t h e

change i n induced a n g l e of a t t a c k c h a r a c t e r i s t i c s p a s t a = 1 6 degrees .

According t o Reference 2 , t h e v o r t e x shed from a n a s p e c t r a t i o 2.0 d e l t a wing

w i l l begin t o b u r s t a t t h e l e a d i n g edge o f t h e t a i l i n t h e 14 t o 16 d e g r e e

a n g l e of a t t a c k range f o r t h e v a r i o u s wing- ta i l s e p a r a t i o n d i s t a n c e s t e s t e d .

Vortex b u r s t i n g can b e s t be desc r ibed a s t h e r a p i d breakdown of a v o r t e x

i n t o random tu rbu lence . Reference 2 i n d i c a t e s t h a t a spec t r a t i o and Mach

number have a s t r o n g i n f l u e n c e on v o r t e x b u r s t i n g . Decreases i n a s p e c t

r a t i o and superson ic Mach numbers teqd t o d e l a y t h e b u r s t i n g o f v o r t i c e s

shed from d e l t a wings. The d a t a a v a i l a b l e were n o t syo temat ic enough t o

show whether o r not v o r t e x b u r s t i n g could be r e l a t e d t o t h e changes i n c.

I n s u f f i c i e n t s u p e r s o n i c d a t a were a v a i l a b l e t o donduct a n a n a l y s i s l i k e

t h a t f o r t h e t r a n s o n i c d a t a . Data from R e f e r e x e 20 were a v a i l a b l e t o produce

v a l u e s of e which compared w i t h t h o s e ob ta ined from t h e t r a n s o n i c d a t a up

t o 72 d e g r e e s a n g l e o f a t t a c k . No superson ic d a t a were a v a i l a b l e t o determine

how E v a r i e d i n t h e 2:' t o 30 degree range. I n t h e t rnn9onic c a s e most of

t h i s region was h igh ly Mach number s e n s i t i v e ; t he r e fo re , use of Figure 152b

f o r angles g r e a t e r than 22 degrees i n the supersonic regime is not advised,

Use of Method

To i l l u s t r a t e t he use of the method f o r p r ed i c t i ng A$ , a general TWV

desc r ip t i on of t he procedure is presented, followed by a step-by-srep

numerical example.

Determine t he d i s t ance , R,, between t he wing t r e i i i n g edge

and t he lead ing edge of t he t a i l a t a l a t e r a l pos i t i on

defined by yw - & 4 ,

Determine t he v e r t i c a l d i s t ance be tween . the vor tex core and

t he t a i l sur face a s a func t ion of a lpha uefng Equation (49).

Using Sect ions 5.1.4 and 5.2.1 c a l c u l a t e C using t he ca lcu la ted %(B)

q u a n t i t i e s C

For angles of a t t a c k t o 16 degrees, use t he resu l t ; of

d I/; , s t e p s 2 and 3 t o c a l c u l a t e $-) . Note t h a t i n t h i s

s t e p S - S re f base'

Using t h e r ,osul ts of s t e p k attd Figure 152a determirie va lues

of t f o r a ~ g l e s of a t t a c k t o 16 degrees,

For t ransonic P r c h nmioers use Figure 152b f o r angles of

a t t a c k between 16 and 30 degreee. (cI6 = E a t a = 16')

~ u p e r s o n i c a l l y , use of Figure 152b t o determine va lues of E

f o r angles of a t t a c k beyond 22 degrees is not advised.

Using Sect ion 5.1.5, c a l c u l a t e CN a s a func t ion of a. T

I

8 Wring the resultlr of Stcc? 5; 5 and 7, determine values' of -

Numerical Example

Calculate A$ a t ll - 1.1 fo r the,body-wing-tail configuration ' TWV

v i t h the following character is t ics .

Body :

L - 10.0 - d * 3.75 inches d

wings:

AR - 2.0 X - 0.0 d - - 0.35 8

%* 12.11 sq. in. , , s - 3.48 inches

CR - C.% inches A ~ , ~ , = 0.

- 15 .40 inches

Ta i l s :

- 7.909 aq. in . s - 2.812 inches s ~ s .P

CR - 5.625 inchee - 'T.E. - 0.

Z.E. - 31.872 inchee

- 1 Calculate g -

- Yw - - "' - 2.733 inches r

2 Calculate h/d a s a function of a

3 Using Section 5 . 1 . 4 ( p . 9 l f f ) and 5.2.1 - (p.143f f) calculate XcB)

d 1 /2 4 Cdcu la te (K) - where Stat base

using r e s u l t s of s t eps 2 and 3.

5 Using the resu1:s of Step 4 and Figure 152a determine - E at a - 16 degrees.

6 Determine e f o r angies of attack between 16 and 30 degrees a t - M-1.1 using Figure 152b. Ut i l i z ing the value of E 'at a = 16

degrees , the-values of t are obtained a t a greater than 16

degrees.

3 using the results of Step 5, 6, and 7 determine AC

Nm*

E . A C N ~ A C N ~ - S . P . D . P .

Data Comparisons

The r e s u l t s of t h e numerical example a r e compared with experimental

d a t a i n Figure 153. The r e s u l t s show good agreement. Fur ther comparisons

a r e presented i n Figures 154, 155 and 156. These l a t t e r f i g u r e s compare

normal fo r ce c o e f f i c i e n t s f o r complete b o d y r i n g - t a i l conf igure t ione w i th

experimental da ta . These p r ed i c t i ons required the use of s eve ra l methods

i n conjunct ion with t he method f o r p red ic t ing AC . A range of Mach N~~~

numbers and configuration geometries were covered and good a g r e d e n t

was obtained i n a l l cases . Figures 155 and 156 represen t independent:

comparisons s i n c e these d a t a were no t used t o development t h e method.

Conf ig. 133

1

Ffgure 150. Transonic Wind Tunnel Teet Configurations

- -- - Vortex Path

- - - - - - T a i l Leadfng Edge Plane

Figure 151. Wing Vortex Location

> ' C Note N ~ ( ~ )

Figure 1 5 2 . Wing Vortex induced Tail Angle of Attack

0 Experimental (Ref. 20)

- Predicted

Fhure 154 . Campartson Retween Predicted And Experimental Results, c ~ R m * W-o.7

0 Experimental. (Ref. 3 4 )

- predicted ( ~ o r ~ e n s & . ' r CN . Ref. 12) B


Figure 155. Comparison Between Predicted And Experimental Results, c ,U-0.85 N~~

0 bxperbntal (Ref. 36)

0 5 10 1 5 20 25


Figure 156 . Comparison B e t w e e n P r e d i c t e d And Experimental Results , , U-2.36

5.4.4 S f f ec t ive Center of Pressure of t he Incremental T a i l Normal

Force Due t o Wingg t --

A method is presented f o r pred ic t ing the e f f e c t i v e center , o f pressure ,

, of the incremental fo rce produced on a ta i l due t o the addit lol l x ~ ~ A n n r of wings t o the body. This force r e s u l t s from the e f f e c t i v e angle of a t t ack ,

E , induced on the t a i l due t o the v o r t i c e s emanating from the wing. The

a v a i l a b l e dat? made i t pogsible t o i d e n t i f y c o r r e l a t i o n s up t o ~ n g l e s of

a t t a c k of 30 degrees i n t ransonic flow and t o approximetely 22 degrees i n

supersonic flow.

Background

The add i t i on of wings t o a b o d y t a i l configurat ion a l t e r s the normal

fo rce produced by t h e t a i l s i n t he presence of a body by an amount i d e n t i f i e d

a s A C N ~ . This incremental normal fo rce l a a t t r i b u t e d t o t he e f f e c t of wing

v o r t i c e s on t h e t a i l . Wing v o r t i c e s produce a change i n the f lowf ie ld

encountered by the t a i l s . The net e f f e c t is t o induce an e f f e c t i v e angle of

a t t a c k on the t a i l s , thereby a:tering the t a i l ' angle of a t t ack . Sect ion 5.4.3

presents a method f o r pred ic t ing the vortex induced angle of a t t ack , r , and

the corresponding value of A Q T W V . TO account f a r the e f f e c t s of A C N ~ on

t o t a l configurat ion cen te r of preasure a method is required t o p red i c t i t s

e f f e c t i v e cent+= of pressure, %pAw.

Method D+velopment

According t o ~ e f t r e n c e 30, 4 p A m can be t rea tod i n a way t ha t is

analogue t o t he e f f e c t of body upwash an the tai l . , i . e , , t h a t both the upwash

and downwash a l t e r t he loads on the t a i l birt do not change the chordwise

d i s t r i b u t i o n appreciably. Therefore, %prim = k p T ( B l = XcpT. 'The

, procedure for' p r e d i c t i q X q is outl ined i n Section 5.1.4 (p. 91 f f ) .

The following exprersion wae ured t o ca lcu la te XCPBw . - t d

Thin aquation require8 the use of a number of the predict ive methods

d e r c r i k d e a r l i e r which w i l l not be r e p u t e d here.

Data Cmparirone - Pigurer 157 and 158 rhow comparieons of the uae of the e f fec t ive center

of prerrure, X C ~ f o r the i n c r m n t e l n o m l force of the t a i l due t o A W

wing vortax f.nterfarance. In there carea the Xcp war used i n an overa l l A W

prediction of the center of premrure, XcpBW, f o r the complete body-wing-tail.

Cornpariaon between the predicted and experimental r e r u l t r a r e good i n both

crmeonic and ruperoonic ra8imer, a t l e a r t f o r the two cares examined.


- Predicted'

10 , 20


Figure 157. Comparison Batveen Predicted And Experimental Data, XCPgm,M-0.85 - d


- Predicted

0 5 1 0 15 20 25


Figure 158. Comparison Bctween Predicted and Experimental Data. ,H=2.36

5.5 Thruet Vector Control Effects

5.5.1 Incremental Nonnal Force Due t o Plume Effects

A method is presented f o r estimating JCN , the incremental normal force co- BP

e f f i c i e n t on a alender t a g c a t ogive-cylinder body due t o a flowing main j e t

(ACNgp). This wthod covers angles of a t t ack 60 180 degrees and a Mach number

range oE 0.60 t o 2.20.

Ba~k.q,:ound

The addit ion of a flowing j e t t o a body produces a change i n the body

trormal force coeff ic ient due t o impingement of the j e t plume on the body and

the e f fec te of the j e t on the flow f i e r d about the body. The magnitude o f t h i s

.?acremental normal force coeff ic ient (ACNBp) is dependent on the following:

Mach number, angle of at tack, and the strength of the j e t r e l a t ive to the f r e e

stream (defined here as the momentum r a t i o MR). No previously derived method

was found which predicted the e f f e c t s of a flowing msin j e t across the desired

angle of a t tack range. The present work describes the formulation of a method

f o r predict ing AGBp up t o a - 180' a t Mach numbers from 0.60 t o 2.20. Data

from t e a t s on a pa r t i cu la r USAP miss i le design form the basis fo r t h i s analyst&.

The incremental normal force coeff ic ient on a body due t o a j e t plume is

defined as:

ACNB~ C N B ~ - CN*

Test da t a were ava i l ab l e a t Mach numbers 0 .60 t o 2.20 and angles of

a t t a ck from 15 t o 165 degrees. J e t momentum r a t i o s tes ted were as follows:

Xach - XR - 0.60 60.1

0.85 30.1

1.20 19.1

1.80 73.6

2 . 2 0 49 .3

Jet--on da ta were ava i lab le only fo r a body-rtrake-tail configure-

t ion , with je t-off da ta obtained on body alone and body-rtrake-tail

cbni igxrat ione. I t would obviously have been des i rab le t o h v a tes ted

the body alone with jet-on. Since t h i s is not ava i l ab l e i t was necessary

2 Assumitq tha t the increment i n normal force , due t o the s t r ake p lus - t a i l , Jei-on, is proport ional t o , t h e increment i n normal fo r ce

coe f f i c i en t , j e t -of f , compute jet-on e t rake p l u s t a i l increment:

t o der ive body alone jet-on normal force coe f f i c i en t s (C ) from ava i lab le N~~

data . The ava i lab le data cons is t of parameters meesured by fn tegra t ix~g

surface@ preeeures, including normel f c r ce coeff ic2ente f o r the body i n t he

presence of s t r akes and t a i l s , jet-on and jet-off ,(C NB(ST)P and CK ); B(ST)

body alone normal force coe f f i c i en t , jet-off (L ); t a i l normal f o r c e N~

coeff Lrient, jet-on and jet-off (C and CN ); and s t r ake normal f u r r e NTP T

coe f f i c i en t , jet-on a r d jet-off (CNSp and C N ~ ) . The procedure used was

as followe:

1 CoLpute the incremental normal force due t o presence of s t r ake - and t a i l with the J e t o f f .

Subt rac t t he ca l cu l a t ed jet-on increment from the measured normal

fo r ce c o e f f i c i e n t of t he body i n t he presence of s t r a k e s and

t a i l s wi th j e t on:

'NBP C N g ( ~ ~ ) ~ 'IB(ST)P

Method Development

Figure 159 s h o w the genera l form of a curve of ACN BP

vereue angle of

a t tack . This curve ahowa t h a t t h e r e is no s i g n i f i c a n t j e t e f f e c t a t ang l e s

of a t t a c k less than 40 degrees. The t e n ACN reachem a peak about a - 70.. BP

then decreases t o a minimum value a t a - 90'. J e t e f f e c t s i nc r ea se aga in a e

alpha approaches 145*, then decrease t o a va lue of zero a t a - 180.. The

value and s ign of ACN a t a - 70' and a - 145' a r e Mach number dependent. BP

A power s e r i e s formulat ion Lncorporatlng t h ~ e f f e c t s of angle of a t t a c k , Mach

number, and momentum r a t i o waa t he approach s e l ec t ed t o f i t a general curve

t o t he da ta , The term ACN was considered t o be l i n e a r l y dependent on j e t BP

momentum r a t i o f o r a given Mach numher.

Power aertea for the variation o t A C N ~ ~ with a . i n w ~ c h h l u e e of zero

occur a t 0- 40., 90" and 180" and valuer of 1.0 occur a t 70" and 145'

vere then constructed, #e form of the equation i a :

- Jet momentum ratio - qj L,

K = Amplification factor - K(M)

A - Power ieries d e f i n i n g curve £ o m

The cmplexlty of the variation with alpha neccasitatca dividing the

mgle of attack range into three interval.: 01 a 5 40.. 400 5 a go0 and

90" 5 a 5 180". Parametare i n each range ere ae follows:

0 < a 5 40' -

A - 23.4450 -89 .88886~~ + 121.1061a2 -66.9524n3

*12.8829a 4 (a 'radians)

'145

A a 45,6283 -82.7i36n+53.6644a2 -14.6512o 3

, +12.8829a 4 ' (a ' radians)

The quantity ACN /%, as determined from the test data. MI non- BP

dirnensiotulized by the value at a = 70. in the range of alpha between 40

and 90 degrees. In the alpha range from 90 to 180 degrees, the value at

a = 145. was used to non-dimensionalire ACN /MR. Figdre 160 ehowe the BP

curve which was faired through the non-dimensionalired test data. The data

for all Mach numbers is combined in arriving at Figure 160. The Mach number

effect ia ,obtained by plotting the values of (ACN /MR) at a = 70°, 145' BP

and then fairing curves through these data to obttin Figure 161.

It rhould be noted that available test data incorporated only one jet

ahmenturn ratio at each t A t Mach number. While these represent realistic

\slues for the configuration tested, eetimates obtained for a missile with

greatly different jet momentum ratios shou1.d be used with caution. Also of

mportance is the fact that the effects of nozzle exit diameter on ACN RY

cmnot be determined from existing data. The ratio of nozzle exit diameter

to body diameter h for the configuration teetted ;as 0.81. It is ) redsonable to assume that this analysis is valid for cases in which the

no:.:le exit diameter approximates that of the body.

Use of ~ethod

The nuthod i m , u t i l i z e d as follows:

Given: a tangen t o g i w - c y l i n d e r body wi th a main j e t ammantun

r a t i o , MR, a t t h e d e s i r e d Mach number.

proceed thus:

1 Determine K - 70 and K145 f o r t h e a p p r o p r i a t e Mach number

(Figure 161)

2 Look up v a l u e s of A f o r t h e d e s i r e d a n g l e s of a t t a c k - (Figure 160)

3 Compute -

where K I K70 f o r 40' a 2 90"

= K f o r 90' 5 a 5 180" 145

Numerical Example

Given t h e fol lowing parameters , compute AC f o r a s l e n d e r tangent 'BP

ogive c y l i n d e r body a t Mach 0.85:

dnor --- 0.90 r e f

1 From Figure 161: KqO - 0.074 -

2 - 3 U t i l i z i n g Figure 160 t o o b t a i n - - v a l u e s of A t h e fol lowing t a b l e is

generated.

Data C o q a t i a o n s - In Figure 162 wethcd r e s u l t s a r e p lo t t ed afong with those da t a used

h formulating t he method. It can be seep t h a t t he curve f i t t i n g

approach used ycelds a good approxim.ltion of AC across t he Mach range. N~~

A lack of independent body alone jet-on da t a a t high angles of a t t a c k

ekes f u r t h e r de t a i l ed comparisons impossible a t t h i s time. Independect

da t a presented i n Referenre 37 f o r a body p lus t a i l conf igura t ion tend t o

support t h i s a n a l y s i s i n t h a t no j e t ' e f f e c t s a r e evident a t angles of

a t t a c k l e s s than about 40 degrees, t he magnitude of AC is small r e l a t i v e t o N~~

t o t a l CN, and the va lue of ACN decreases with increasing Mach number. BP

80 120


Figure 159: General C u r v e Form, AC Nw

Figure 161. Amplification Pactore for Calculating ACN BP

e) Mach 2.2'

v Teat Data - Estimate

- -


Figure 142 (Cont.). Comparisons Between Predictions And Experimental Data,

5.5.2 Effective Center of Pressure f o r Incremental Body Normal Force Due

t o Plume Effects

Summary

Amethod is presenied for esii.ating XCp , the c f f c c t h e Cenrfr @ f Pressure BP

of the incremental force on e slender tangcut ogive-cylinder body due to

a f l a r i n g main j e t . This method app l i e s f o r angles of a t t ack t o 180 degrees

and a Mach number range of 0.60 t o 2.20.

Background

, The addit ion of a flowing mein j e t t o a body produces a change i n the

body center of preeeure locat ion due t o plurre impingement on the body and

plume in te rac t ion with the flowfield about the body. No methods were found

t o predic t the c a t e r of pressure locat ion over the desired high angle of

a t t a c k range. The present work describes the formulation of such a method.

The data 'forming the bas i s f o r t h i s corre la t ion were obtained from t e s t s on

a pa r t i cu la r USAF missi le design.

Test data were avai lable a t Mach numbers 0.60 t o 2.20 and angles of

a t t ack f r m 15 t o 165 degrees. J e t momentum r a t i o s tes ted were a s follows:

Uach - Pk - 0.60 60.1

,O. 85 30.1

1.20 19.1

1.80 73.6

2.20 49.3

For the configuration tested. &= 2.5 and = 14.45. d d ,

Jet-on data were ~ v a i l a b l e only f o r a body-strake-tail configura-

t ion , with jet-off data obtained on body alone and body-strake-tail

coafigurations. It was therefore necessary t o derive XCpae using a v a i l a b l l

data. Speci f ica l ly the quan t i t i e r obtained exparimentally vere measured

by in tegra t ing preorure d ie t r ibu t ionr and consisted of no-1 force

coeff ic ient8 and centere of pressure of the body alone (jet-off) , of

the atrakea and ta i l s ,and the body i n the preseace of the s t rakee and ' t a i l s

(jet-on and jet-off) . The der ivat ion of jet-on values of the incremental

CN on the body due t o etrake and t a i l carryover (IB(ST)p) is described i n

the mechod presented f o r determining A C N ~ . The procedure developed t o

ca lcula te XcpRp 18 rr followe.

Method Developrant

Figure 163 showa the bas ic data used i n formulating the jet-on center

of presaure predict ion method. Due t o a lack of t e s t data i n which body '

, ,

fineness r a t i o ( l /d) was varied, i t wae decided t o b a s e the predict ion

method on jet-off values of XcpB which may be calculated using the method

of SecFion 5.1.2.

Examination of the data i n Fjgure 163 reveals tha t the flowing main ,

j e t has e ssen t i a l ly no e f fec t on the body center of pressure lacat ion a t

angles of a t tack l e s s than about 100 degreas a t a l l Mach numbers. A t PI-1.2

and below, XCpgp f a l l s about 0.5 ca l ibe r s forward of XcpB fo r 10O0( n 2 160°.

A t supersonic Mach numbers, XC% and XcpBp a r e essen t i a l ly equal up to

a - 120°, then a re symmetrical about the value a t a = 120'.

The method developed simply approximates the cuntes of Figure 163 a s

described above. For ~%ch nu bars l e s s than or equal t o 1.2:

lI)oO< a 110': Linearly in terpola te between valcos a t a = 100' and l l o O

For Mach aumberro greater than 1.2:

0.1 u 2 120' : ICPw/d * XmB/d '

1 , x a 3 - 2 K 1 - - d

X CPB

where K1 - value of - d a t a = 120'

Usa of Xejthod-

The method is used as follows:

Given s s?.eder tangent

The jet momentrun rat io % is

test data pre*ziously cited.

ogive-cylinder body with a flo*.ng main jet.

of the same order of,magnitude as thase of the

Proceed thusly: X * ~

1 Determine -r for the Mach and e l p h s range desired from test data or - via the mettrod of Sectio? 5.1.2 Cp. 61 f f ) .

x r n ~ X~ 2 If M 5 1.2, - - for 0 5 a 2 loOD d d

x~Qp %pa If M > 1.2, - -- d d for 0 2 a 120'

X %P 'CP~

If M > 1.2 , -7- 2 K, -- d for 1209 5 a 5 180°

' C P ~ where K1 - value of ,, @ a = 120' d

Numerical Exau~~& -- ktauine X /d for,a slender tangent ~give-cylinder body at Mach

CPBP

0.85 and Mach 1.8; % - 30.1 qt Mach 0.85, MR - 73.6 at Mach 1.8.

Mach 0.85:

(Step I)

(teat data)

(Step 1)

Data Comparisons

I n Figure 164, method r e s u l t s a r e compared w i th t h e jet-on d a t a used

i n f o r m u l a t i q the method. I t can be seen t h a t t h i s r e l a t i v e l y simple

method y i e l d s a good approximation of X /d ac ros s Mach number and angle of CPBP

a t t a c k regime. A l ack of independent body a lone je t-on da t a a t t he necessary

high angles of a t t a c k . u k e s f u r t h e r d e t a i l e d comparioons impossible a t

t h i a time. Independent d a t a presented i n Reference 37 f o r a body p l u s t a i l

conf igura t ion i n d i c a t e t rends s i m i l a r t o those noted i n t h i a ana ly s i s , i . e . ,

l i t t l e j e t a f f e c t on XCp a t anglee of a t t a c k l e s s than 100 degrees, than a B

f o w a r d s h i f t f n CP loca t i on ; r i t h increas ing angle of a t t a ck .

F i g u r e

60.6

0' J e t Off

v ~ e t OII I l a

ANCLE OF ATTACK-DEG.

b) Uach 0.85

ANCLE OF ATTACK-DEC.

6 3 . C o m p a r i s o n O f Body Alone X (.Tet On V e r a u s Jet O f f ) 2 d

0 40 80 120 160

' ANGLE OF ATTACK-DEG.


Figure 163 (Cone,.). Comparisou Of Body Alone XCp (Jet On Versus J e t O f f ) ' -

0 ' 40 80 , , 1 2 0 160

, ANGLE OF ATTACK-DEG.

F i g ~ i r ~ 163 (Cont .) . Comparison O f Body Alone XCp (;et On Versus Jet O f f )

0 40 80 120 160


b) Mach 0.85

- Estimate

MGLE OF ATTACK-DEG.

Figure 164. Comparison Between ~ r e d i = t i o n s And Experimental Dar.a. - d


I I d ) Mach 1.80 I 0 Test Data

- Estimate I ' I I

Figure 164 (Cont.). Comparison Between Predictions and Experimental Data,

d

' v Teat Data - Estbate

80 120


5.5.3 Incremental T a i l Normal Force Due To Plume Ef fec t s

' A method i e , p r e s e n t e d t o p r e d i c t AC , t h e incremental normal fo rcc

c o e f f i c i e n t on hor izonta l tai ls on a slende; tangent ogive-cylinder body

due t o a jet pLne . The term A% r ep re sen t s t he change i n t o z a l n o r m 1

fo rce coe=f i c i en t on two ta i l panels p lu s the change i n tail-on-body carryover

normal fo rce due t o a flowfng jet. The method is appl icable a t angles of

a t t a c k up t o 180 degrees a t Mach numbers 0.60 t o 2.20.

Background

The add i t i on of a flowing jet t o a bod; . . ta i l conf igbra t ion produces

changes i n t he nonaal fo rce on the t a i l and ?-I t he magnitude of the carryover

normal fo rce imposed on t h e body by t h e t a i l s . The magnitude of t h i s j e t

e f f e c t ia dependent.on such parameters a s angle of a t t a c k , Mach number, tat.!

s i z e , and the s t r eng th of t h e jet r e l a l i v e t o t ' , e f r e e streaci (defined here

a e the momentum r a t i o , %). NQ previous ly deri*.,t>.i method was found which

predic ted the a f f e c t s of a flowing j e t on the t a i l s a t the des i red high

angles of atLack. The present work descr ibes the fo rnh la t i on of such a

method f o r p red i c t i ng a t angles of ac tack up t o 180 degrees and Mach

numbere 0.60 t o 2.20. Data from t e s t s on a , p a r t i c u l a r USAF m i s s i l e conf,igo-

r a t i o n form t h e bas i e f o r c h i s ana lys i s .

The incremental normal fo rce c o e f f i c i e n t on a body-tail conf igura t ion due

t o jet e f f e c t s on the t a i l i s deftned a$:

T h i u is baaed on the premise t h a t t he t o t a l e f f e c t of a t a i l on a body-

t a i l conf igura t ion is made up of t h e fo rce on the t a i l i t s e l f p lu s the ca r ry

over t o the body, and f u r t h e r t h a t both q u a n t i t i e s may be a f f ec t ed by the

preranca of a plume.

Test da t a were ava i l ab le a t Mach numbers 0.60 t o 2.2 and angles of

a t t a c k from 15 t o 165 degrees. J e t momentum r a t i o s t e s t e d . m r e a s follows:

J e t -on da t a were ava i l ab le only for a body-strake-tail conf igura t ion ,

with je t-off da t a obtained on body a lone and body-strake-tai ls configurat ions.

It was, t he re fo re , neceesary t o d e t i v e AC f r m ava i l ab le data. Parameters NTP

mcasured by i n t e g r a t i n g su r f ace pressures included normal force coe f f i c i en t s

f o r the body i n the presence of s t r akee and t a i l s , je t-off and jet-on ( B i W and CN ; body a lone normal fo rce c o e f f i c i e n t , j e t -of f (C 1; t o t i 1 t a i l

N~ c o e f f i c i e n t i n the presence of t he body, jet-of £ and jet-on

and CN ; and t o t a l r t r a k s normal fo rce c o e f f i c i e n t ,

jkt-off and jet-on f p and CN . The procedure f o r computing

S(B)Pto ta l

AC from known da ta is a s follows : N~~

Given the bas i c equation

The terms CN and CN may be determined d i r e c t l y from t e s t

T (B)P to ta l T(B)tota:,

data.

The q u t i o n may then be expressed as

- CW * - C .+ AIBTpe

(B)Ptot*l Lir(B) t o t a l

The incremental normal fo rce c o e f f i c i e n t s , IB(ST) and IB(ST)p, due t o

presence of s t r a k e and ta i l , were developed previous ly i n Sec t ion 5.5.1

One can then de f ine

I "B(ST)P I IB(ST)P - %(ST) (52)

It was assumed t h a t t he changes i n t a i l carryover on the body due t o t he j e t

would be ' p ropor t i ona l t o t h a t f o r a s t r a k e p lus t a i l i n t he same rmnner a s t he

change i n no'rmal fo rce on t h e t a i l due t o a j e t Lo propor t iona l t o t h a t for

a s t :ake p lus t a i l . Therefore: AC

AIB(T)p b ,ST) P N~ (B (53)

A% (B) AC

N~ (B)

The r e s u l t s of eqoat ion (53) may then be s u b s t i t u t e d i n t o equat ion (51)

determine ACx . TP

Method Development

Figure 165 shows the genera l form of curves of I A C versus angle of N~~

a t t a c k f o r 0.6 2 M 2 1.2 and 1.2 < M L 2.2. Both curves show no l e t e f f e c t s

a t angles of a t t a c k l e s s thari 20 degrees, followed by increas ing I A C

V8ch numbers t e s t ed occur a t a = 1100 and a = '1600, while zero po in t s f a l l a;

,J * 135' f o r M 2 1.2 and a t a = 120' f o r 1.2 < H 2 2.2. AC =duals zero a t N~~

.+ = 180' a t a l l Mach numbers due t o symmetry. The value and s i g n cf AC N~~

a = 55'. 119'. and 16G0 a r e Mach number dependeit.. A power s e r i e s f o r m u l a ~ i o n

incorpora t ing the e f f e c t s of n n ~ l c of a t t a c k , Mach number. j e t momentum r a t i o ,

and t a i l a r ea was t he approach se i ec t ed t o f i t a genera l cqlrve t o the d a t t .

Asauming ACN v a r i e s l i n e a r l y wi th %and RT, then 7 2

1 where I$ - t a i l a r e a r a t i o = - .

'ref

A(a) is defined by the genera l curve f o q ~ i n Figure 165, and v a r i e s i n

magnitude from zero t o one. The magnitude of ~ ( a ) is scaled by the values of

(AC /M#,$ a t a = 5S0, 110' and 160'. respec t ive ly , i n the t h r ee *TP

I ranges of angle of a t t a c k . Power s e r i e s curves with a v a l l e of zero a t

a = 20°, 90°, 135'. and 180' and a value of 1.0 a t a = 55", 110°, and 160'

were then constructed f o r ~ & h c u b & l e s s than or equal t o 1.20. Curves

constructed f o r M > 1.2 had zero va lues a t a = 90°, 120' and 180.. The

f i n a l form of t he equat ion is:

ACum MR * RT * K * A '

where MR = j e t momentu. r a t i o - q~/q, ,

RT - t a i l a r ea r a t i o - ?q/Sref

K = Bmplif icat ion f a c t o r

= K55 f o r 0' ( a ( 90' (0.6 ( M ( 2.2)

f o r 90' < u ( 135O (0.6 ( M ( 1.2)

for 90' < a ( 120' (1.2 < M ( 2 . 2 )

f o r 13 .5 '~ a 5 180' (0.6 5 M 5 1.2)

f o r 120' < a ( 180" (1.2 < M 2.2)

A = Power s e r i e s def in ing curve form.

Solution of the equation necessitates dividing the angle of a t tack

range in to four in tervals . For Mach nmbers from 0.60 to 1 . 2 :

0 2 a 5 20"

A = 0 :. ~ C N ~ ~ = 0

20" 5 a 5 90"

= K55

A = 2.4498 - 16.7515a + 37.3514a * - 29.8329a 3 + 7.77403

[a ' radians)

90' La: 135'

110

A - -256.1760 + 494. 79370-359.0i37a2 +116.7324a3 -14.38370 4

[a r r a d ians 1

135' la: 180'

K -K 160 3 A 1046.9190 -l5Ol.628Oa+ 798.0073a2 -185.0520~ + 16,0493a 4

[a +radians I For 1.2 4 4 2 2.2:

0 :a2 20"

A - 0 ACN- - 0

20" :a2 90"

-%5 3 A = 2.4498 -16.751%+ 37 .3514a2 -29.8329~ + 7.7740~ 4

[a c r a d i a n s ]

Values of k55r Ri18# and K160 have been determined empir ical ly and a r e p lo t ted

versus Uach number i n Figure 166. Plwe- s e r i e s A is presented versus angle of

a t t a c k i n Figure 167a f o r Ml 1 .2 and i n r*igure i i 7 b f o r 1.2 < M 5 2.2.

It should be noted t h a t a v a i l a b l e t e s t da t a incorporated only one j e t

momentum r a t i o a t each Mach numbe~. 'While these represent r e a l i s t i c values

f o r t he conf igura t ion t e s t e d , es t imates obtained for a mi s s i l e with g rea t ly

d i f f e r e n t j e t momentum r a t i o s should be used w i t h caut ion. Also of

importance is t h e f a c t C , h t t h e e f f e c t s of varying nozzle e x i t diameter and

nozzle- to- tai l d i s t ance cancot be derived from e x i s t i n g da ta . The r a t i o of

nczzle e x i t diameter t o body dlamater (dnoz/dref) f o r t h e donfigurat ioq

t e s t ed was 0 .8 ; ; t he d i s t ance from tho nozzle e x i t plane t o the t a i l t r a i l i n g

edge wm 0.42d. Var ia t ion of these parametera can be expected t o have

rams as y e t undetermined effects on t h e va lues of ACN predicted by this , TP

met hod.

Use of Method

The method is used a s follows:

Given a tangent ogive-cylinder with ho r i zon ta l tails of a r ea r a t i o ,

I$, and a main jet momentum r a t i o , MR.

Proceed thus :

1 ~ e t e r m i n e K55, KllO, and K160 f o r t he des i red Xach number. - ((Figure 166)

2 Look up values of A f o r t he des i red angles of a t t a c k i n t he - appropr ia te Mach range. (Figure 167)

3 Compute A C N ~ ~ = MR * RT * K * A - where K = liS5 f o r 0 2 a 5 90"

= Kl10 f o r 90" 2 a 5 135" i f M 5 1.2

= Kl10 f o r 90" < a 2 120" i f 1.2 < M < 2.2 - Kl60 f o r 135" < a 2 180" i f M < 1.2 -

= K160 f o r 120' < a 5 180" i f 1.2 < M < 2.2 - Numerical Example

Given the following parameters, compute ACNTp f o r a s lender tangent

ogive-cylinder body with hor izonta l t a i l s from a = 0 to a = 180" a t Hach 0.85

and Mach 1.80.

MR 30.1 a t M = 0.85

MR = 73.6 a t M = 1.80

RT = 0.8b

A t h c h 0.85:

1 From Figure 166: - K55 ' 0.065, Kl10 -0.028, Ki6* = 0.015

A * MR * RT * K A C N ~ ~

(Figure 167a) - 0 . 30.1 0 .84 0.065 0

Data Compariaons

A * l+ * * K AQTp

(Fig. 167b)

In Figure 168, method r e s u l t 8 are p l o t t e d along with those data used i n

fozmulating the -4thod. The c u r v e - f i t t i n g approach used y i e l d s a good ap-

proximetion o f ACNTp across the Mach range t e s t e d . A lack o f independent data

makes further canparisoncl impossible sf t h i e time.

40 80 120 160

M G L E OF ATTACK-DEG.

Figure 165. General Curve Forms, I ^"nP I

UACH NUMBER

0 0.4 0.8 1.2 ' 1.6 2.0

MACH NUMBER

Figure 166. Amplification Factors for Calculating ACN TP

MACH NUMBER Figure 166 (Cont.). Amplification Factors for Calculating dC

N~~

e) 'Mach 2.20

0 Teat Data - Prediction , % " 4 9 . 3 4

ANGLZ OF ATTACK-DEG.

Figure 168 (~ont.). Compsrisono Between Predictions And Experimental Data , AC NTP

5.5 .4 Effective Center of Pressure of Incremental Tail Normral Force Due

to Plume Effects

Summary

A method is presented for predicting X , the effective center of C P ~ ( ~ ) ~

pressure of the incremental tail normal force due to plume effects. Data

comparisons showed no difference between jet-on and jet-off tail chordwise

center of pressure for Mach numbers between 0.6 and 2.2 and angles of attack

to 180 degrees. 'i :?refore, it is not necessary to develop a netg method and

it is recornended that the existing method of Sec ion 5.1.5 be used to

calculate XCp /CR which is equivalent to XCp T

F R * T(B)P

Background

When predicting :he aerodynamic characteristics f ~ r a missile at high

angles of attack, the presence of an exhaust plume must be taken into account.

At high angles of attac.k, the plume prpduced by a thrusting m?.esile can alter

local surface pressures through either direct impingement or by its influence

on the flowfield forward of the plume. Methode for predicting plurne effects

on body nonnal force, body center of pressure, and td.1. normal force have

been presentcd in Sections 5.5.1 through 5.5.3. This section deals specif<-

cally with the effects of a plume on tail chordwise center of pressure.

A study has been completed on the effects which rocket motor exhaust

pLumes have on tail center of pressure. Data used in the sttldy were obcnincd

iron wind tunnel tests of a particular USAF body-strake-tail missile -nnf i ~ u -

ratjon. Tests vere canducted using a pressure model wi t5 and vitlio1.r~ :wic

jet sinulation at Mach numbers from 0.6 to 2.2 and ang!es of ~s.ttsct ' r c n

15 -0 :A'; dtqqrecs. The ratio af jet total pressure to free s t r e m t -ca l

pressure and tb* ratio of jet dynamic pressure to free srrea I E::zmic DreswfF

and

and

0 Jet-On

c) n - 1.2

--- ANGLE OF ATTACK-DEG.

Figure 169 (Cant.) Coqarison Between Jet-On and Jet-Off Tail Centers of Pressure

6.0 CONCLUSIONS AND RECOMMENDATIONS

This study shows that the availability of systematic test data permits

the development of methodology to predict reasonably accurate aerodynamic

characteristics. The applicability of the methods is limited only by the

range of the test data. As for any semi-empirical method, the methods should

not be used beyond the range of the test data base until the real limits

df applicability can be ascertained. This can only be accomplished over a

period of time as additional test data becomes available.

Experience gained in using the methods shows that although they err

suitable for "hand" calculations, it is desirable to computer+ze them.

This was not included as part of the present contract, and is therefore

recommended for future consideration.

The succesu of the methods developed here supports the view that this

approach could well be extended as the systuaatic data base grows. Areas

which were identified as deficient in data or as Fertile ground for the

continuation of the effort begun here are summarized below:

1 Since wind tunnel testing does not, in general, match flight - Reynolds numbers and Mach numbers simultaneously, this causes

8 question about the accuracy with 3hich Reynolds number

effects can be accounted for in the methods. T5is uncertainty

manifests itself primsrily in the modeling of the viscous

contribution to the body normal force. Additional tests

aimed specifically at assessing the vircoum effects on body

normal force are reconnnended.

2 Since maneuverability implies the ure of a control syetem, - the tast data base and methods should now be extended to

dea l wi th def lec ted c o n t r o l sur faces .

3 Cer ta in goometric f e a t u r e s , e .g . , b o a t t a i l s and nose - bluntness,should a l s o be t e s t ed sys temat ica l ly t o

complement the (Vlrrent da t a base.

4 The e f f e c t s of a r b i t r a r y r o l l angle should be t r ea t ed - systeinat ical ly beyond angles of a t t a c k of 45 degrees

which was t r ea t ed i n the recent Martin Mariet ta s tudy

(Reference 38) conducted f o r t he U. S, Army. One of

the problem a r e a s of p a r t i c u l a r i n t e r e s t t n t h i s regard

?s the p red i c t i on of hinge moments on the l ee s ide , '

aurfaces even a t small angles of a t t a c k wherein the

occurrence of couples complicates t he pr?dic t ion of

t he cen t e r of pressure on the t a i l ,

5 Final?.y i t should be r e c a l l e d t h a t t h i s s tudy d e a l t - only v f t h s t a t i c aerodynamics, whereas s i m i l a r methods

can and should be developed f o r some of t he dynamic

e t e b i l i t y der iva t ive@.

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Craves , E. B . , "Supersonic Aerodynamic ~ h a r a c t c r i s t ' i c . ~ of a Low-',spec: R a t i o M i s s i l e Model w i t h Wing and T a i l C o n t r o l s and w i t h T a i l s in Line and I n t e r d i g i t a t e d " NASA 'I'M;Y-25'JIv 1972. ,

C a r t e r , S. K., e t a 1 "Aerodynamic C h a r a c t e r l s t i z s o f a ? t i s s i l r Conf iguratLon i n t h e Presence o f an Exhaust Pli!me ~t ~ n ~ l e s of ~ t t i 1 c . k t o 180 Degrees (U) , I ' MnAC Paper WD 2521, .Julv 1 9 7 5 .

A i ~ l l o , G. F. , "AFROPYNAMIC METHOI)OI,OC.Y, Hodics wi th . \ rh i t r , i r \ Rnl' Angles ( ' t ransonic and Supersonic) ," I\R 14,145. F i t ~ ~ l l Kepvt t on I 1 . S . Army MICOM C o n t r a c t F13615-75-C--7052, 19?6.

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